Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.21 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 101 2168 2835033 995162 2026-09-29T11:50:00Z Sompisintuli 3111827 /* School president */ new section 2835033 wikitext text/x-wiki {{Schooltalk}} ==What's Next== Edit this area as a to-do list bulletin board of topics that should be either discussed or implimented on the School's portal. *Discuss Site Redesign :While this new enhanced format is quite attractive, at such a stage of progress where much of the initial layout of this school has yet to be fully realized, it makes it quite difficult for an interested party to contribute to this page. Instead of only taking a moment to edit the page, one must search through the code of the wiki to find specifically where they wish to add to, subtract from or alter. If possible, the school should have a draft page that it simply a blueprint for a final page with an impressive layout such as this that will eventually become the Wikiverity School of Law homepage. *Separate topics into introductory, intermediate, and advanced. :Many of the main topics already created build upon other areas which are more basic. Such areas such as property, contract, procedure, torts, legal writing, etc. should not be overlooked. *Create a topic regarding Law School :include discussion of experiences, concerns, lwa review, moot court, focuses, goals, etc. *Provide a section related to pertinent, timely issues :will alter the future for law, such as the development of new areas, problems the law are struggling with, issues of new impression, etc. *Implement Priestly 11. *Determine how to deal with country specific and jurisdiction specific law. ==Subject Areas== I am an attorney and would like to participate in building Wikiversity School of Law. I suggest that Internet Access Law be made a discrete subject area of the curriculum. There are numerous fraud, privacy and safety issues that are being addressed in the courts of the United States. The same may be true for other countries where internet use is widespread. As internet access becomes more global, these issues will proliferate and laws will be enacted to address them. [[User:jstewart]][[User:Jstewartesq|Jstewartesq]] 15:56, 5 March 2007 (UTC) ==Subject Areas== Hi. This is my first real contribution to the project. Hooray! Well, here goes: From the outset, there needs to be the distinction made between the British [[w:Common law|Common law]] system and [[w:Civil law|Civil law]] systems. Perhaps we could use [[w:Australia|Australia]]'s [[w:Priestley 11|Priestley 11]] to decide what subjects should be party of our Law School. The priestley 11 includes: * [[w:Criminal law|Criminal law]] and procedure; * [[w:Torts|Torts]]; * [[w:Contract|Contract]]s; * [[w:Property law|Property]] both real (including Torrens System Land) and Personal; * [[w:Equity|Equity]] (including [[w:Trust law|Trusts]]); * [[w:Administrative law|Administrative law]]; * Federal and State [[w:Constitutional law|Constitutional law]] ([[w:Australian constitutional law|Australian]]; [[w:British constitutional law|British]]; [[w:United States constitutional law|U.S.]]); * [[w:Civil procedure|Civil Procedure]]; * [[w:Evidence (law)|Evidence]]; * Professional Conduct (including basic trust accounting) ([[w:American Bar Association Model Rules of Professional Conduct|American Bar Rules of Conduct]]; * [[w:Corporations law|Company law]]. In addition to these subjects, it is also important that students of law study [[w:Jurisprudence|Jurisprudence]], [[w:International law|International law]], [[w:Customary law|Customary law]] and the [[w:Court|court system]]. Cheers, [[User:Jpeob|Jpe]]|[[User talk:Jpeob|ob]] 23:56, 25 August 2006 (UTC) ===Another one:=== How about Patent and/or Copyright law? Would it be localized to each country? It makes sense to me to have a lesson on international IP law, but national IP laws are certainly important/broad enough to have courses (lessons?) on. Perhaps int'l law would be a portion of each nat'l course? --[[User:AdamG|AdamG]] 20:03, 28 September 2006 (UTC) :My take on this is to make sure it is known as you are entering the topic which country/jurisdiction is covered. This will prevent any confusion and will also promote creation of brand new pages for different countries and jurisdictions instead of editing existing pages. This is one area that frustrates me now in Wikipedia and I would hate to see it happen here if we can prevent it. What do you think? -[[User:Ramjaw|Ramjaw]] 13:37, 3 October 2006 (UTC) === Traffic Law, contracts and more === Im interested in traffic law, and some other forms. You can find some course outlines here at the bottom of the page http://www.senecac.on.ca/fulltime/LCA.html http://www.senecac.on.ca/fulltime/CTA.html http://www.senecac.on.ca/fulltime/LCKA.html ===Modular Subjects=== I would think that an approach of modularising subjects would work well instead of a tangled mess? * Legal System - How it all works explains basicas of law developes to different systems in different counrtries: includes common law, primary legislation, court system, professions within the system etc... * Criminal Law - everything specific to criminal law obviously Obviously many more modules can be thought of. ===Added Jurisprudence=== I added Jurisprudence as a department area. As soon as exams are over (next two weeks) I will put up some kind of beginner and intermediate lessons for jurisprudence. Any suggestions appreciated. --[[User:JHessick|JHessick]] 23:16, 12 April 2007 (UTC) == School page redesign == I want to make it look like [http://en.wikiversity.org/w/index.php?title=School:Language_and_Literature&oldid=29950] this. Hopefully I can get this done today. --[[User:AdamG|AdamG]] 22:10, 29 September 2006 (UTC) :Got the bulk of it done. I'm still going through the images at MediaWiki to find better images, but that can be done incrementally. Have fun.--[[User:AdamG|AdamG]] 22:51, 29 September 2006 (UTC) ::Nice job! You can find many nice images and icons at [[:commons:|Wikimedia Commons]]. When using an image from the Commons, you don't have to upload it here, just write its name on the page and it will appear. :) Did you know that the first Serbian constitution was the Constitution of the Medieval Serbian Empire, written on the State Council of Skopje in 1349, during the regime of the Emperor Stefan Dušan? It was one of the oldest constitutions in the world. And the first modern Serbian constitution was brought on the 15th of October, 1835. It was quite democratic for that times, so under the preasure of Russia and Austria, which did not have constitution themselves back then, it was rejected. :) --[[User:Ђорђе Д. Божовић|George D. Bozovic]] ^{[[User talk:Ђорђе Д. Божовић|talk]]} 12:15, 30 September 2006 (UTC) :::Oh yeah, I meant Commons :P. I guess I just can't keep track of 12-odd projects!--[[User:AdamG|AdamG]] 13:09, 30 September 2006 (UTC) ::Love the site redesign but I am concerned about accessability of the average user/contributor. I have a much more difficult time adding simple edits and at this stage I imagine there will be tremendous overhaul of the main page to create a respectable presentation. -[[User:Ramjaw|Ramjaw]] 04:49, 3 October 2006 (UTC) I am interested in family law and all the different areas this type of law offers [[User:Nancysen13|Nancysen13]] 18:36, 18 January 2007 (UTC) == Legal Writing == Since I know little about coding in wiki style or namespaces, etc. I have not done much in the way of contributions; I would however be willing to assist in the legal writing area if no one else is interested. [[User:GP|GP]] 16:54, 11 February 2007 (UTC) == Structure == Maybe it will be wise to elevate Civil Law, Common Law, International Law, etc to Centers. And within each Center a department of Public and Private Law. Particular national jurisdictions could have departments which would link to general concepts. [[User:Carlos.ivan.fuentes|C.I.]] 19:52, 1 March 2007 (UTC) == I'd like to volunteer == I'm a non practicing lawyer from Montreal, Quebec, Canada. Quebec is somewhat unique in the fact that it's the only jurisdiction I'm aware with that exists as a sort of hybrid system, with certain fields of law being covered by the Civil Law tradition, while other matters are dealt with accoring to the Common Law. I posess both a Common Law degree (LLB) as well as a Civil Law Degree (BCL). In fact, McGill took the bold and innovative step in creating what they call a transsystemic program. For example, Civil Law Contracts and Common Law Contracts are taught simultaneously, by the same professor, in the same lecture. Though it would be my pleasure to discuss any legal issue, I'd especially recommend any who may be curious as to the differences between the Common Law and the Civil Law to request any clarification they may require. [[User:Loomis|Loomis]] 00:29, 6 April 2007 (UTC) Quebec is not that unique; Louisiana has a similar civil law/common law hybrid system. {{unsigned|68.13.220.58|05:02, 26 November 2007}} == link - 'Is an unsecured FTP server publicly accessible?' == [http://www.cnet.com/8301-13796_1-9909415-79.html?tag=bnpr 'Is an unsecured FTP server publicly accessible?'] - May be of interest... --[[User:Remi|Remi]] 06:43, 4 April 2008 (UTC) == Structure of the School - Jurisdictional == Hi,<br /> I have only just started looking at contributing to Wikiversity, particularly as a law student. The issue I have is the somewhat tangled approach currently. Would it not make more sense to organise the school based on jurisdictions, as a starting point? I appreciate this is extremely over-simplistic as, for example, English and Welsh law is intrinsically linked to USA law, EU law and Scots law. If we can get a consensus on this, the next stage would be to plan how it would work, and then put it into place.<br /> The alternative is to have various jurisdictions on the same page, which would be extremely confusing to most users.<br /> Thanks, [[User:Jhfireboy|Jhfireboy]] ([[User talk:Jhfireboy|talk]]) 23:42, 24 January 2013 (UTC). : I tend to agree with you. Most current and former English Empire nations would have a similar approach to law, where nations like China, Japan, Russia and others, even France and Germany, may view law very differently, if I've understood your idea. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|talk]]) 01:01, 25 January 2013 (UTC) ::Exactly. I think that if the School considered each national law (jurisdiction) separately, and has other pages that show the differences, then it would be easier for a student of a particular nation to understand their own law. It also allows them to view laws of other nations separately a be able to distinguish the differences more readily. As you say, the law of France is very different from English law; as the French use a civil law system and the English uses a common law system. 12:37, 25 January 2013 (UTC) :::Go for it! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|talk]]) 15:46, 25 January 2013 (UTC) == School president == someone to assist in managing school Assist the principal about school safty And needs [[User:Sompisintuli|Sompisintuli]] ([[User talk:Sompisintuli|discuss]] • [[Special:Contributions/Sompisintuli|contribs]]) 11:50, 29 September 2026 (UTC) 1o4krjq5kd22vloxjpq6vpxujurpsq4 0 37372 2834858 2834473 2026-09-28T14:48:05Z Evgenii Arsentev 3110906 Add sub-page link 2834858 wikitext text/x-wiki {{AI_navbar_0}} {{RightTOC}} Artificial intelligence involves intelligence that is created through computers or other means other than humanness. This area is for learning, teaching, and research related to artificial intelligence. AI usage has evolved into [[agentic harnesses]] and more. ==Learning resources== * [[Artificial neural network]] * [[Cross-domain AI topics]] ===Sub-pages === * [[/Introduction/]] * [[Artificial intelligence/prospects|Prospects for artificial intelligence]] ** [[Supersymmetry/String Theory based Artificial Intelligence]] * [[/Lecture aid/]] * [[/AI content engine economics/]] * [[/Checking AI output with a second agent/]] * [[/Giving an AI agent access safely/]] * [[/Checking an open-source AI project before you depend on it/]] ==Discussion questions and essay ideas== * Can artificial intelligence (AI) be used as a tool for good? List applications scenarios of AI. Discuss ethical aspects that are related to decision making and decision support with AI. Define requirements and constraints of AI application that you could derive from your analysis. * Identify areas of application where AI that can assist staff members e.g. in health care facilities to get more time for social interaction and support for patients? What happens if AI applications fail and patients get harmed? What level of security must be applied and who is responsible for that. Apply the underlying concept on other topics like autonomous driving, ...) * Analyse the concept of [[machine learning]] and analysis of big data! What is the importance (PRO and CONs) of AI in this context in comparision to other mathematical or computer science methodologies? * Assume AI is used for pattern recognition in social media data or in general in the context of [[Commercial Data Harvesting]]. Provide examples and analyse the possible consequence of false-positive resp. false negative classification of individuals! What would be your response and what would you suggest to improve the considered system? * How can the human brain or human mind be emulated using computers? Is it desirable to? * What other ethical issues are involved with artificial intelligence? * How can artificial intelligence be utilized and created in an ethical way? === More Discussion questions and essay ideas=== * Compare prompt engineering to loop engineering to graph engineering. ** Compare and contrast these different modalities. ** When and how can each one be useful? * What are some different notable agentic harnesses and what the pros and cons of some of the major ones? ==Educational/research/learning prompt ideas== * Create pages for an educational wiki that demonstrate how agentic AI can be utilized to build out an educational wiki. * What goals should an educational wiki have. That is, for a wiki meant for learning, teaching, and research, how should such collaborative content actually benefit the readers and collaborators. * How could AI generated video be utilized for learning, teaching, and research? * What are economic aspects of universities that readers and creators on distributed educational wikis can possibly learn from or benefit from? * How those who utilize agentic AI benefit from agentic AI utilize decentralized wikis to have their agent collaborate on their behalf and on behalf of humanity? * How could Hermes agents collaborate to advance educational wikis - wikis that are meant for learning, teaching, and research? === Wikipedia === ==== Wikipedia Readings/Pages ==== {{colbegin|4}} : {{w|Explainable artificial intelligence}} : {{w|Artificial intelligence}} : {{w|Accelerating change}} : {{w|Quantum computing}} : {{w|Artificial brain}} : {{w|Artificial general intelligence}} : {{w|Mind uploading}} : {{w|Technological singularity}} : {{w|Intelligence explosion}} : {{w|Superintelligence}} : {{w|AI takeover}} : {{w|Existential risk from artificial general intelligence}} : {{w|Friendly artificial intelligence}} : {{w|Computronium}} : {{w|Distributed artificial intelligence}} : {{w|Machine learning}} : {{w|Philosophy of artificial intelligence}} : {{w|Hardware for artificial intelligence}} : {{w|Weak AI}} : {{w|Deep learning}} : {{w|History of artificial intelligence}} : {{w|Synthetic intelligence}} : {{w|Template:Natural language processing}} (template) : {{w|List of datasets for machine-learning research}} : {{w|Fine-tuning (machine learning)}} : {{w|Convolutional neural network}} : {{w|Natural language processing}} : {{w|Transformer (machine learning model)}} : {{w|Prompt engineering}} : {{w|PaLM}} : {{w|GPT-4}} : {{w|Auto-GPT}} : {{w|BLOOM (language model)}} : {{w|Hugging Face}} : {{w|Hallucination (artificial intelligence)}} : {{w|AI safety}} : {{w|AI alignment}} : {{w|Existential risk from artificial general intelligence}} : {{w|Template:Existential risk from artificial intelligence}} (template) : {{w|Ethics of artificial intelligence}} : {{w|Generative artificial intelligence}} : {{w|Artificial general intelligence}} : {{w|Artificial intelligence art}} : {{w|Music and artificial intelligence}} : {{w|Generative pre-trained transformer}} : {{w|Regulation of artificial intelligence}} : {{w|Wikipedia:Large language models}} (policy) : {{w|Language model}} : {{w|LLaMA}} {{colend}} ===== Wikipedia Categories ===== : {{wc|Artificial intelligence}} :: {{wc|Artificial neural networks}} ::: {{wc|Neural network architectures}} :: {{wc|Machine learning}} ::: {{wc|Datasets in machine learning|... datasets}} ::: {{wc|Deep learning}} :::: {{wc|Deep learning software|... software}} :: {{wc|Generative artificial intelligence|Generative AI}} :: {{wc|Applications of artificial intelligence}} ::: {{wc|Large language models}} ::: {{wc|Natural language processing}} ::: {{wc|Speech recognition}} :: {{wc|Existential risk from artificial general intelligence}} :: [[w:List of artificial intelligence projects|AI projects]] ===External === * [https://www.youtube.com/watch?v=Jy3zjXK4ao4 Google Just Put an A.I. Brain in a Robot (Research Breakthrough)] (video) ==See also== [[Image:Artificial Intelligence & AI & Machine Learning - 30212411048.jpg|thumb|300px|right]] {{colbegin|3}} * [[Robotics Projects]] * [[Introduction to Robotics]] * [[Arduino Robot]] * [[Automation]] * [[History of artificial intelligence]] * [[Anthropomorphic Robotics Project]] * [[Large language models]] * [[Robot car]] * [[Automatic Control and Robotics]] * [[Open-source domestic robots]] * [[Portal:Artificial intelligence]] * [[Center_for_Corporate_Auditing,_Responsibility_and_Management_policy_Authoring/Artificial_intelligence_eschewal_policy|Artificial intelligence eschewal policy]] * [[Wikiversity:Artificial_intelligence]] (Policy) {{colend}} ==External links== * [https://www.elementsofai.com/ Course on Elements of AI] * [https://www.reddit.com/r/artificial/ Discuss artificial intelligence on Reddit] * https://tldr.tech/api/latest/ai [[Category:Artificial intelligence| ]] qj7khht65u1s1j24vdd7bs8xvlohi8k 5 56067 2834862 2777877 2026-09-28T15:56:10Z ~2026-52129-14 3111763 2834862 wikitext text/x-wiki __TOC__ == New concept (April 2008667677676767676 drizzzzzyyyyy AHHHHHH!!!!!) == ''The following is a copy of my post to the colloquium.'' --[[User:McCormack|McCormack]] 07:10, 14 April 2008 (UTC) :I've been working on a new browse page for Wikiversity for some time. I have activated this today. If you were used to the old one, the novelty will probably shock. You can still access the old one if you prefer it. The new browse page has a number of ideas behind it. (1) It makes heavy use of the '''category tree''' extension to list wikiversity content dynamically via the category system, which means the page pretty well maintains itself. (2) It uses '''multiple paths and metaphors''' which cater better for the needs of our diverse user types and which avoid any stereotyping of wikiversity (e.g. only for US universities, only for certain resource types, etc). A self-criticism would be that the page feels rather like a train-station now - but then educational institutions do actually need some notice boards which feel like train-stations. Other pages, such as portals, can adopt friendlier, decorative and lounge-like styles. Hopefully people will feel that this is an improvement. --[[User:McCormack|McCormack]] 07:07, 14 April 2008 (UTC) == How to edit this page == ''Simple instructions''. # Don't edit the page; edit the categories! # OK, go ahead and edit the page '''''IF''''' you really know how the category tree extension works and how a few other rather complicated templates work...! You can also ask me ([[User:McCormack|McCormack]]) if you have to do some maintenance on the page. # Suppose you want to add something into the system, here's how... ## Let's say your prize resource is called [[The Telephone Directory Reading Group of Vladivostok]]. ## Start by going to [[The Telephone Directory Reading Group of Vladivostok]] and add the page to some categories. For example, this is a reading group, so add it to [[:Category:Reading groups]]. It is also non-formal education, so add it to [[:Category:Non-formal Education]]. You could also add it to [[:Category:Telecommunications]] and [[:Category:Russia]]. ## Assuming, at this point, that all the above categories are themselves properly categorised into the Wikiversity main category tree, your resource will now be attached to multiple guides. For example, it would show up under "resource types --> reading groups" and "non-formal education", as well as somewhere in the faculty of information sciences. ## Not only would your categorisation add your reading group to this browse page; it would also, at the same time, with no more clicking, add it to the new portals which also use the category tree extension. In other words, at the mere cost of proper and thorough categorisation, your reading group has become well advertised and well connected. ## Suppose that [[:category:Reading groups]] did not exist yet, or was not properly categorised, you would create it, and then categorise it under [[:Category:Resources by type]] to attach it to the overall tree. :: [[User:McCormack|McCormack]] 13:25, 19 April 2008 (UTC) ::<small>Please paste questions below; please do not break up the instructions. Thank you!</small> == "Old" style subject school icons == Although the current version is definatly rather powerful in its functionality, the following is a suggestion for the next version. Maybe include a ''row'' a bit akin to how [[:Template:Portal_Nav]] is setup between the top two boxes, or between the second box and where the columns begin. And while the current version is quite nice, there maybe something to be said for the icons and the simplicity. --[[User:Remi|Remi]] 16:09, 23 May 2008 (UTC) : I had a good look at the [[Wikiversity:Browse]] page again and tried to imagine how it would look if it was somehow combined with [[:Template:Portal_Nav]]. I couldn't really imagine it very well. One option might therefore be for you to create a concept page where you try it out. Another thought I have is that we might be dealing with fundamentally different purposes here, which might be better served by having two quite different pages. If a page tries to please too many needs at the same time, it may end up pleasing none. In this context, one of my long term goals would be to improve [[Help:Guides]] to the point where this can also go in the sidebar under "browse". Perhaps we can also develop a highly iconic navigation page to go in the sidebar along with these? --[[User:McCormack|McCormack]] 04:54, 24 May 2008 (UTC) :They can be seen [[Wikiversity:Browse/Old|here]]. [[User:Emesee|Emesee]] 21:13, 1 September 2008 (UTC) == Resource by name and resource by type == How would it be if we removed the "resource by name", put "resource by type" in its place, and then had the departments right below the "Major starting points"? [[User:Emesee|Emesee]] 03:04, 7 September 2008 (UTC) * Darklama's "resource by name" section is a good idea, but it does take up a lot of horizontal space. Rather than "removing" it, perhaps we should just "move" it to a linked subpage - e.g. keep the box, but replace the box content with a hyperlink to the subpage? A second suggestion is that some of the very small boxes could go at the top. --[[User:McCormack|McCormack]] 04:26, 7 September 2008 (UTC) ** I moved the "resource by name" section to a subpage and linked it from the box at the top quite prominently. I haven't done anything about your layout ideas yet, Emesee, but you might like to experiment. --[[User:McCormack|McCormack]] 10:06, 10 September 2008 (UTC) == removing the top box == Let's remove the top box, and put the link to [[Wikiversity:Browse/Old]] somehow in with the headline text ("Major starting points - click here to browse all Portals") on the box below it. [[User:Emesee|Emesee]] 19:41, 8 September 2008 (UTC) * I think the top box is a useful opportunity to orientate users in the use of the page. The wording needs to be brief and appropriate, though. General purposes: tell people what the page is; explain the concept of multiple pathways; explain to people how to get their pages listed. The last of these points is quite important. Although it's actually very easy to get listed (just categorise your pages properly), some editors, including more experienced ones, haven't realised this over the last few months. So a word of help is good as well. --[[User:McCormack|McCormack]] 10:10, 10 September 2008 (UTC) == Browse concept == [[Wikiversity:Browse/Concept]] [[User:Emesee|Emesee]] 22:02, 20 September 2008 (UTC) == Departments == This list is getting quite long. Perhaps we could consider something slightly more heirarchical? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 11:22, 2 November 2008 (UTC) :I think we should merge Schools, Faculties, Departments and Subjects. These categories overlap and should be simplified first, than we could categories from most general to more specific and just list the [[b:Subject:Major Subjects|major subject]] categories like Wikibooks does. --<span style="font: bold 10pt 'courier new', comic, sans, ms;">[[User:Darklama|{{font|color=midnightblue|dark}}]][[Image:Yin yang.svg|12px]][[User_talk:Darklama|{{font|color=green|lama}}]]</span> 19:31, 3 November 2008 (UTC) ::Merging all those might be too much. But perhaps Departments could simply be merged into Schools and Faculties? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 08:30, 4 November 2008 (UTC) == Overwhelming == I haven't used the "browse" page in quite some time, as I have most of the content I fiddle with watchlisted. I really like the idea of the dynamic lists -- they are powerful and self maintaining. However, they don't work well here. The page is already HUGE, and it will continue to grow (hopefully) as things are added and categorized. Browse should be a good way to quickly and simply identify where you want to go. Now, I'm not suggesting that we take out all of the dynamic lists. But, I know when I first clicked on the page, I was completely lost. It is just so much information, it's hard to take in. The old page, by contrast, while not so simply self-maintaining, was much more compact. I think this actually made browsing itself easier, as you could find your way through the page in a simpler way. The icons were also helpful. Maybe a compromise, where we put out some icons with a few dynamic lists, and perhaps have pointers to other more detailed lists? [[User:Historybuff|Historybuff]] 08:28, 14 February 2009 (UTC) :It was a big mistake to turn the browse page over to automation. Providing a user-friendly guide to Wikiversity content is a job that requires constant community participation. Categorization of pages is just a first step. Every major category should have a user-friendly directory that serves as a guide to the contents of that category. Following the arbitrary convention established long ago at Wikipedia these directories are called [[Wikiversity:Portal|portals]]. Members of the community should devote some of their time to special organizational projects where participants collaborate to organize Wikiversity contents. The major [[Wikiversity:Content development|content development projects]] that support such collaboration are the [[Wikiversity:Schools|Schools]]. --[[User:JWSchmidt|JWSchmidt]] 14:37, 14 February 2009 (UTC) {| style="float: right;" | {{Robelbox|theme=1|title=By Department}} <div style="{{Robelbox/pad}}; font-size:90%" style="height:300px; overflow:auto;"> {{#categorytree:Departments|mode=all|depth=10|hideroot|namespaces=topic category|hideprefix=always}} </div> {{Robelbox/close}} |} ::I am testing this: ::* the ability to limit the size by using a lift at the side, this is 300px, but could be 1000px for departments ::* drill down into sub categories. One sub category that could really be useful would be [[:Category:Department stubs]] for stubs. ::[[User:HenkvD|HenkvD]] 19:58, 27 February 2009 (UTC) == NICE PAGE! == NICE PAGE!!!!!!!! == Extension requested == Hi, 1. I would like to design a course and add content that is MULTIDISCIPLINARY. Can you include that as a category? 2. I am not quite sure of the other aspects of categorisation either. Like school subject (it is not a school subject), or university faculties (could be linguistics, computing or psychology), or the level, or type of resource, because it is being written with a user in mind whose interest lies in solvoing several problems in education. See my intro under [[reflective thinking]], which is just one term asociated with my course planned. --[[User:Genezistan|Genezistan]] 03:45, 13 September 2009 (UTC) :You can actually include your course under ''all'' applicable categories: there is no need to limit yourself! [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 17:26, 28 September 2009 (UTC) Thanks Adviser [[User:Genezistan|Genezistan]] 03:13, 29 September 2009 (UTC) == Delete school of Accountancy == School of Accounting should be deleted form the list of schools altogether since it doesn't exist.<br /> --[[User:Newmangledmatt|ttam]] ([[User talk:Newmangledmatt|discuss]] • [[Special:Contributions/Newmangledmatt|contribs]]) 00:16, 19 October 2014 (UTC) :It's already gone, but you have to purge the page to see the change. If you want to add the purge feature to your list of 'More', go to Preferences, Gadgets and add the Purge tab. Content like this page that is generated from category lists may not update for some time (hours). If you purge, it is updated immediately. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:43, 19 October 2014 (UTC) 15wxdz4tvlnm7l8uzjny8zipf2egicm 2834863 2834862 2026-09-28T15:56:55Z ~2026-52129-14 3111763 2834863 wikitext text/x-wiki __TOC__ == New concept (April 2008667677676767676 drizzzzzyyyyy AHHHHHH!!!!!) == ''The following is a copy of my post to the colloquium.'' --[[User:McCormack|McCormack]] 07:10, 14 April 2008 (UTC) :I've been working on a new browse page for Wikiversity for some time. I have activated this today. If you were used to the old one, the novelty will probably shock. You can still access the old one if you prefer it. The new browse page has a number of ideas behind it. (1) It makes heavy use of the '''category tree''' extension to list wikiversity content dynamically via the category system, which means the page pretty well maintains itself. (2) It uses '''multiple paths and metaphors''' which cater better for the needs of our diverse user types and which avoid any stereotyping of wikiversity (e.g. only for US universities, only for certain resource types, etc). A self-criticism would be that the page feels rather like a train-station now - but then educational institutions do actually need some notice boards which feel like train-stations. Other pages, such as portals, can adopt friendlier, decorative and lounge-like styles. Hopefully people will feel that this is an improvement. --[[User:McCormack|McCormack]] 07:07, 14 April 2008 (UTC) == How to edit this page ((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}} == ''Simple instructions''. # Don't edit the page; edit the categories! # OK, go ahead and edit the page '''''IF''''' you really know how the category tree extension works and how a few other rather complicated templates work...! You can also ask me ([[User:McCormack|McCormack]]) if you have to do some maintenance on the page. # Suppose you want to add something into the system, here's how... ## Let's say your prize resource is called [[The Telephone Directory Reading Group of Vladivostok]]. ## Start by going to [[The Telephone Directory Reading Group of Vladivostok]] and add the page to some categories. For example, this is a reading group, so add it to [[:Category:Reading groups]]. It is also non-formal education, so add it to [[:Category:Non-formal Education]]. You could also add it to [[:Category:Telecommunications]] and [[:Category:Russia]]. ## Assuming, at this point, that all the above categories are themselves properly categorised into the Wikiversity main category tree, your resource will now be attached to multiple guides. For example, it would show up under "resource types --> reading groups" and "non-formal education", as well as somewhere in the faculty of information sciences. ## Not only would your categorisation add your reading group to this browse page; it would also, at the same time, with no more clicking, add it to the new portals which also use the category tree extension. In other words, at the mere cost of proper and thorough categorisation, your reading group has become well advertised and well connected. ## Suppose that [[:category:Reading groups]] did not exist yet, or was not properly categorised, you would create it, and then categorise it under [[:Category:Resources by type]] to attach it to the overall tree. :: [[User:McCormack|McCormack]] 13:25, 19 April 2008 (UTC) ::<small>Please paste questions below; please do not break up the instructions. Thank you!</small> == "Old" style subject school icons == Although the current version is definatly rather powerful in its functionality, the following is a suggestion for the next version. Maybe include a ''row'' a bit akin to how [[:Template:Portal_Nav]] is setup between the top two boxes, or between the second box and where the columns begin. And while the current version is quite nice, there maybe something to be said for the icons and the simplicity. --[[User:Remi|Remi]] 16:09, 23 May 2008 (UTC) : I had a good look at the [[Wikiversity:Browse]] page again and tried to imagine how it would look if it was somehow combined with [[:Template:Portal_Nav]]. I couldn't really imagine it very well. One option might therefore be for you to create a concept page where you try it out. Another thought I have is that we might be dealing with fundamentally different purposes here, which might be better served by having two quite different pages. If a page tries to please too many needs at the same time, it may end up pleasing none. In this context, one of my long term goals would be to improve [[Help:Guides]] to the point where this can also go in the sidebar under "browse". Perhaps we can also develop a highly iconic navigation page to go in the sidebar along with these? --[[User:McCormack|McCormack]] 04:54, 24 May 2008 (UTC) :They can be seen [[Wikiversity:Browse/Old|here]]. [[User:Emesee|Emesee]] 21:13, 1 September 2008 (UTC) == Resource by name and resource by type == How would it be if we removed the "resource by name", put "resource by type" in its place, and then had the departments right below the "Major starting points"? [[User:Emesee|Emesee]] 03:04, 7 September 2008 (UTC) * Darklama's "resource by name" section is a good idea, but it does take up a lot of horizontal space. Rather than "removing" it, perhaps we should just "move" it to a linked subpage - e.g. keep the box, but replace the box content with a hyperlink to the subpage? A second suggestion is that some of the very small boxes could go at the top. --[[User:McCormack|McCormack]] 04:26, 7 September 2008 (UTC) ** I moved the "resource by name" section to a subpage and linked it from the box at the top quite prominently. I haven't done anything about your layout ideas yet, Emesee, but you might like to experiment. --[[User:McCormack|McCormack]] 10:06, 10 September 2008 (UTC) == removing the top box == Let's remove the top box, and put the link to [[Wikiversity:Browse/Old]] somehow in with the headline text ("Major starting points - click here to browse all Portals") on the box below it. [[User:Emesee|Emesee]] 19:41, 8 September 2008 (UTC) * I think the top box is a useful opportunity to orientate users in the use of the page. The wording needs to be brief and appropriate, though. General purposes: tell people what the page is; explain the concept of multiple pathways; explain to people how to get their pages listed. The last of these points is quite important. Although it's actually very easy to get listed (just categorise your pages properly), some editors, including more experienced ones, haven't realised this over the last few months. So a word of help is good as well. --[[User:McCormack|McCormack]] 10:10, 10 September 2008 (UTC) == Browse concept == [[Wikiversity:Browse/Concept]] [[User:Emesee|Emesee]] 22:02, 20 September 2008 (UTC) == Departments == This list is getting quite long. Perhaps we could consider something slightly more heirarchical? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 11:22, 2 November 2008 (UTC) :I think we should merge Schools, Faculties, Departments and Subjects. These categories overlap and should be simplified first, than we could categories from most general to more specific and just list the [[b:Subject:Major Subjects|major subject]] categories like Wikibooks does. --<span style="font: bold 10pt 'courier new', comic, sans, ms;">[[User:Darklama|{{font|color=midnightblue|dark}}]][[Image:Yin yang.svg|12px]][[User_talk:Darklama|{{font|color=green|lama}}]]</span> 19:31, 3 November 2008 (UTC) ::Merging all those might be too much. But perhaps Departments could simply be merged into Schools and Faculties? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 08:30, 4 November 2008 (UTC) == Overwhelming == I haven't used the "browse" page in quite some time, as I have most of the content I fiddle with watchlisted. I really like the idea of the dynamic lists -- they are powerful and self maintaining. However, they don't work well here. The page is already HUGE, and it will continue to grow (hopefully) as things are added and categorized. Browse should be a good way to quickly and simply identify where you want to go. Now, I'm not suggesting that we take out all of the dynamic lists. But, I know when I first clicked on the page, I was completely lost. It is just so much information, it's hard to take in. The old page, by contrast, while not so simply self-maintaining, was much more compact. I think this actually made browsing itself easier, as you could find your way through the page in a simpler way. The icons were also helpful. Maybe a compromise, where we put out some icons with a few dynamic lists, and perhaps have pointers to other more detailed lists? [[User:Historybuff|Historybuff]] 08:28, 14 February 2009 (UTC) :It was a big mistake to turn the browse page over to automation. Providing a user-friendly guide to Wikiversity content is a job that requires constant community participation. Categorization of pages is just a first step. Every major category should have a user-friendly directory that serves as a guide to the contents of that category. Following the arbitrary convention established long ago at Wikipedia these directories are called [[Wikiversity:Portal|portals]]. Members of the community should devote some of their time to special organizational projects where participants collaborate to organize Wikiversity contents. The major [[Wikiversity:Content development|content development projects]] that support such collaboration are the [[Wikiversity:Schools|Schools]]. --[[User:JWSchmidt|JWSchmidt]] 14:37, 14 February 2009 (UTC) {| style="float: right;" | {{Robelbox|theme=1|title=By Department}} <div style="{{Robelbox/pad}}; font-size:90%" style="height:300px; overflow:auto;"> {{#categorytree:Departments|mode=all|depth=10|hideroot|namespaces=topic category|hideprefix=always}} </div> {{Robelbox/close}} |} ::I am testing this: ::* the ability to limit the size by using a lift at the side, this is 300px, but could be 1000px for departments ::* drill down into sub categories. One sub category that could really be useful would be [[:Category:Department stubs]] for stubs. ::[[User:HenkvD|HenkvD]] 19:58, 27 February 2009 (UTC) == NICE PAGE! == NICE PAGE!!!!!!!! == Extension requested == Hi, 1. I would like to design a course and add content that is MULTIDISCIPLINARY. Can you include that as a category? 2. I am not quite sure of the other aspects of categorisation either. Like school subject (it is not a school subject), or university faculties (could be linguistics, computing or psychology), or the level, or type of resource, because it is being written with a user in mind whose interest lies in solvoing several problems in education. See my intro under [[reflective thinking]], which is just one term asociated with my course planned. --[[User:Genezistan|Genezistan]] 03:45, 13 September 2009 (UTC) :You can actually include your course under ''all'' applicable categories: there is no need to limit yourself! [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 17:26, 28 September 2009 (UTC) Thanks Adviser [[User:Genezistan|Genezistan]] 03:13, 29 September 2009 (UTC) == Delete school of Accountancy == School of Accounting should be deleted form the list of schools altogether since it doesn't exist.<br /> --[[User:Newmangledmatt|ttam]] ([[User talk:Newmangledmatt|discuss]] • [[Special:Contributions/Newmangledmatt|contribs]]) 00:16, 19 October 2014 (UTC) :It's already gone, but you have to purge the page to see the change. If you want to add the purge feature to your list of 'More', go to Preferences, Gadgets and add the Purge tab. Content like this page that is generated from category lists may not update for some time (hours). If you purge, it is updated immediately. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:43, 19 October 2014 (UTC) ltnaidr4ygvmoy11cwi2tzzf6xgpsa1 2834864 2834863 2026-09-28T15:57:14Z ~2026-52129-14 3111763 2834864 wikitext text/x-wiki __TOC__ == New concept (April 2008667677676767676 drizzzzzyyyyy AHHHHHH!!!!!) == ''The following is a copy of my post to the colloquium.'' --[[User:McCormack|McCormack]] 07:10, 14 April 2008 (UTC) :I've been working on a new browse page for Wikiversity for some time. I have activated this today. If you were used to the old one, the novelty will probably shock. You can still access the old one if you prefer it. The new browse page has a number of ideas behind it. (1) It makes heavy use of the '''category tree''' extension to list wikiversity content dynamically via the category system, which means the page pretty well maintains itself. (2) It uses '''multiple paths and metaphors''' which cater better for the needs of our diverse user types and which avoid any stereotyping of wikiversity (e.g. only for US universities, only for certain resource types, etc). A self-criticism would be that the page feels rather like a train-station now - but then educational institutions do actually need some notice boards which feel like train-stations. Other pages, such as portals, can adopt friendlier, decorative and lounge-like styles. Hopefully people will feel that this is an improvement. --[[User:McCormack|McCormack]] 07:07, 14 April 2008 (UTC) == How to edit this page <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> <div class="user-block" style="background:#ffe0e0; color:var(--color-base-fixed,#202122); border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px"> [[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code> <hr> If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div> ((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}}((subst:blocked}} == ''Simple instructions''. # Don't edit the page; edit the categories! # OK, go ahead and edit the page '''''IF''''' you really know how the category tree extension works and how a few other rather complicated templates work...! You can also ask me ([[User:McCormack|McCormack]]) if you have to do some maintenance on the page. # Suppose you want to add something into the system, here's how... ## Let's say your prize resource is called [[The Telephone Directory Reading Group of Vladivostok]]. ## Start by going to [[The Telephone Directory Reading Group of Vladivostok]] and add the page to some categories. For example, this is a reading group, so add it to [[:Category:Reading groups]]. It is also non-formal education, so add it to [[:Category:Non-formal Education]]. You could also add it to [[:Category:Telecommunications]] and [[:Category:Russia]]. ## Assuming, at this point, that all the above categories are themselves properly categorised into the Wikiversity main category tree, your resource will now be attached to multiple guides. For example, it would show up under "resource types --> reading groups" and "non-formal education", as well as somewhere in the faculty of information sciences. ## Not only would your categorisation add your reading group to this browse page; it would also, at the same time, with no more clicking, add it to the new portals which also use the category tree extension. In other words, at the mere cost of proper and thorough categorisation, your reading group has become well advertised and well connected. ## Suppose that [[:category:Reading groups]] did not exist yet, or was not properly categorised, you would create it, and then categorise it under [[:Category:Resources by type]] to attach it to the overall tree. :: [[User:McCormack|McCormack]] 13:25, 19 April 2008 (UTC) ::<small>Please paste questions below; please do not break up the instructions. Thank you!</small> == "Old" style subject school icons == Although the current version is definatly rather powerful in its functionality, the following is a suggestion for the next version. Maybe include a ''row'' a bit akin to how [[:Template:Portal_Nav]] is setup between the top two boxes, or between the second box and where the columns begin. And while the current version is quite nice, there maybe something to be said for the icons and the simplicity. --[[User:Remi|Remi]] 16:09, 23 May 2008 (UTC) : I had a good look at the [[Wikiversity:Browse]] page again and tried to imagine how it would look if it was somehow combined with [[:Template:Portal_Nav]]. I couldn't really imagine it very well. One option might therefore be for you to create a concept page where you try it out. Another thought I have is that we might be dealing with fundamentally different purposes here, which might be better served by having two quite different pages. If a page tries to please too many needs at the same time, it may end up pleasing none. In this context, one of my long term goals would be to improve [[Help:Guides]] to the point where this can also go in the sidebar under "browse". Perhaps we can also develop a highly iconic navigation page to go in the sidebar along with these? --[[User:McCormack|McCormack]] 04:54, 24 May 2008 (UTC) :They can be seen [[Wikiversity:Browse/Old|here]]. [[User:Emesee|Emesee]] 21:13, 1 September 2008 (UTC) == Resource by name and resource by type == How would it be if we removed the "resource by name", put "resource by type" in its place, and then had the departments right below the "Major starting points"? [[User:Emesee|Emesee]] 03:04, 7 September 2008 (UTC) * Darklama's "resource by name" section is a good idea, but it does take up a lot of horizontal space. Rather than "removing" it, perhaps we should just "move" it to a linked subpage - e.g. keep the box, but replace the box content with a hyperlink to the subpage? A second suggestion is that some of the very small boxes could go at the top. --[[User:McCormack|McCormack]] 04:26, 7 September 2008 (UTC) ** I moved the "resource by name" section to a subpage and linked it from the box at the top quite prominently. I haven't done anything about your layout ideas yet, Emesee, but you might like to experiment. --[[User:McCormack|McCormack]] 10:06, 10 September 2008 (UTC) == removing the top box == Let's remove the top box, and put the link to [[Wikiversity:Browse/Old]] somehow in with the headline text ("Major starting points - click here to browse all Portals") on the box below it. [[User:Emesee|Emesee]] 19:41, 8 September 2008 (UTC) * I think the top box is a useful opportunity to orientate users in the use of the page. The wording needs to be brief and appropriate, though. General purposes: tell people what the page is; explain the concept of multiple pathways; explain to people how to get their pages listed. The last of these points is quite important. Although it's actually very easy to get listed (just categorise your pages properly), some editors, including more experienced ones, haven't realised this over the last few months. So a word of help is good as well. --[[User:McCormack|McCormack]] 10:10, 10 September 2008 (UTC) == Browse concept == [[Wikiversity:Browse/Concept]] [[User:Emesee|Emesee]] 22:02, 20 September 2008 (UTC) == Departments == This list is getting quite long. Perhaps we could consider something slightly more heirarchical? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 11:22, 2 November 2008 (UTC) :I think we should merge Schools, Faculties, Departments and Subjects. These categories overlap and should be simplified first, than we could categories from most general to more specific and just list the [[b:Subject:Major Subjects|major subject]] categories like Wikibooks does. --<span style="font: bold 10pt 'courier new', comic, sans, ms;">[[User:Darklama|{{font|color=midnightblue|dark}}]][[Image:Yin yang.svg|12px]][[User_talk:Darklama|{{font|color=green|lama}}]]</span> 19:31, 3 November 2008 (UTC) ::Merging all those might be too much. But perhaps Departments could simply be merged into Schools and Faculties? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 08:30, 4 November 2008 (UTC) == Overwhelming == I haven't used the "browse" page in quite some time, as I have most of the content I fiddle with watchlisted. I really like the idea of the dynamic lists -- they are powerful and self maintaining. However, they don't work well here. The page is already HUGE, and it will continue to grow (hopefully) as things are added and categorized. Browse should be a good way to quickly and simply identify where you want to go. Now, I'm not suggesting that we take out all of the dynamic lists. But, I know when I first clicked on the page, I was completely lost. It is just so much information, it's hard to take in. The old page, by contrast, while not so simply self-maintaining, was much more compact. I think this actually made browsing itself easier, as you could find your way through the page in a simpler way. The icons were also helpful. Maybe a compromise, where we put out some icons with a few dynamic lists, and perhaps have pointers to other more detailed lists? [[User:Historybuff|Historybuff]] 08:28, 14 February 2009 (UTC) :It was a big mistake to turn the browse page over to automation. Providing a user-friendly guide to Wikiversity content is a job that requires constant community participation. Categorization of pages is just a first step. Every major category should have a user-friendly directory that serves as a guide to the contents of that category. Following the arbitrary convention established long ago at Wikipedia these directories are called [[Wikiversity:Portal|portals]]. Members of the community should devote some of their time to special organizational projects where participants collaborate to organize Wikiversity contents. The major [[Wikiversity:Content development|content development projects]] that support such collaboration are the [[Wikiversity:Schools|Schools]]. --[[User:JWSchmidt|JWSchmidt]] 14:37, 14 February 2009 (UTC) {| style="float: right;" | {{Robelbox|theme=1|title=By Department}} <div style="{{Robelbox/pad}}; font-size:90%" style="height:300px; overflow:auto;"> {{#categorytree:Departments|mode=all|depth=10|hideroot|namespaces=topic category|hideprefix=always}} </div> {{Robelbox/close}} |} ::I am testing this: ::* the ability to limit the size by using a lift at the side, this is 300px, but could be 1000px for departments ::* drill down into sub categories. One sub category that could really be useful would be [[:Category:Department stubs]] for stubs. ::[[User:HenkvD|HenkvD]] 19:58, 27 February 2009 (UTC) == NICE PAGE! == NICE PAGE!!!!!!!! == Extension requested == Hi, 1. I would like to design a course and add content that is MULTIDISCIPLINARY. Can you include that as a category? 2. I am not quite sure of the other aspects of categorisation either. Like school subject (it is not a school subject), or university faculties (could be linguistics, computing or psychology), or the level, or type of resource, because it is being written with a user in mind whose interest lies in solvoing several problems in education. See my intro under [[reflective thinking]], which is just one term asociated with my course planned. --[[User:Genezistan|Genezistan]] 03:45, 13 September 2009 (UTC) :You can actually include your course under ''all'' applicable categories: there is no need to limit yourself! [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 17:26, 28 September 2009 (UTC) Thanks Adviser [[User:Genezistan|Genezistan]] 03:13, 29 September 2009 (UTC) == Delete school of Accountancy == School of Accounting should be deleted form the list of schools altogether since it doesn't exist.<br /> --[[User:Newmangledmatt|ttam]] ([[User talk:Newmangledmatt|discuss]] • [[Special:Contributions/Newmangledmatt|contribs]]) 00:16, 19 October 2014 (UTC) :It's already gone, but you have to purge the page to see the change. If you want to add the purge feature to your list of 'More', go to Preferences, Gadgets and add the Purge tab. Content like this page that is generated from category lists may not update for some time (hours). If you purge, it is updated immediately. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:43, 19 October 2014 (UTC) mjan3tr6spdnyrxem6atiegs7g6xnt6 2834865 2834864 2026-09-28T15:57:37Z NDG 3006541 Reverted edits by [[Special:Contributions/~2026-52129-14|~2026-52129-14]] ([[User_talk:~2026-52129-14|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]] 1872968 wikitext text/x-wiki __TOC__ == New concept (April 2008) == ''The following is a copy of my post to the colloquium.'' --[[User:McCormack|McCormack]] 07:10, 14 April 2008 (UTC) :I've been working on a new browse page for Wikiversity for some time. I have activated this today. If you were used to the old one, the novelty will probably shock. You can still access the old one if you prefer it. The new browse page has a number of ideas behind it. (1) It makes heavy use of the '''category tree''' extension to list wikiversity content dynamically via the category system, which means the page pretty well maintains itself. (2) It uses '''multiple paths and metaphors''' which cater better for the needs of our diverse user types and which avoid any stereotyping of wikiversity (e.g. only for US universities, only for certain resource types, etc). A self-criticism would be that the page feels rather like a train-station now - but then educational institutions do actually need some notice boards which feel like train-stations. Other pages, such as portals, can adopt friendlier, decorative and lounge-like styles. Hopefully people will feel that this is an improvement. --[[User:McCormack|McCormack]] 07:07, 14 April 2008 (UTC) == How to edit this page == ''Simple instructions''. # Don't edit the page; edit the categories! # OK, go ahead and edit the page '''''IF''''' you really know how the category tree extension works and how a few other rather complicated templates work...! You can also ask me ([[User:McCormack|McCormack]]) if you have to do some maintenance on the page. # Suppose you want to add something into the system, here's how... ## Let's say your prize resource is called [[The Telephone Directory Reading Group of Vladivostok]]. ## Start by going to [[The Telephone Directory Reading Group of Vladivostok]] and add the page to some categories. For example, this is a reading group, so add it to [[:Category:Reading groups]]. It is also non-formal education, so add it to [[:Category:Non-formal Education]]. You could also add it to [[:Category:Telecommunications]] and [[:Category:Russia]]. ## Assuming, at this point, that all the above categories are themselves properly categorised into the Wikiversity main category tree, your resource will now be attached to multiple guides. For example, it would show up under "resource types --> reading groups" and "non-formal education", as well as somewhere in the faculty of information sciences. ## Not only would your categorisation add your reading group to this browse page; it would also, at the same time, with no more clicking, add it to the new portals which also use the category tree extension. In other words, at the mere cost of proper and thorough categorisation, your reading group has become well advertised and well connected. ## Suppose that [[:category:Reading groups]] did not exist yet, or was not properly categorised, you would create it, and then categorise it under [[:Category:Resources by type]] to attach it to the overall tree. :: [[User:McCormack|McCormack]] 13:25, 19 April 2008 (UTC) ::<small>Please paste questions below; please do not break up the instructions. Thank you!</small> == "Old" style subject school icons == Although the current version is definatly rather powerful in its functionality, the following is a suggestion for the next version. Maybe include a ''row'' a bit akin to how [[:Template:Portal_Nav]] is setup between the top two boxes, or between the second box and where the columns begin. And while the current version is quite nice, there maybe something to be said for the icons and the simplicity. --[[User:Remi|Remi]] 16:09, 23 May 2008 (UTC) : I had a good look at the [[Wikiversity:Browse]] page again and tried to imagine how it would look if it was somehow combined with [[:Template:Portal_Nav]]. I couldn't really imagine it very well. One option might therefore be for you to create a concept page where you try it out. Another thought I have is that we might be dealing with fundamentally different purposes here, which might be better served by having two quite different pages. If a page tries to please too many needs at the same time, it may end up pleasing none. In this context, one of my long term goals would be to improve [[Help:Guides]] to the point where this can also go in the sidebar under "browse". Perhaps we can also develop a highly iconic navigation page to go in the sidebar along with these? --[[User:McCormack|McCormack]] 04:54, 24 May 2008 (UTC) :They can be seen [[Wikiversity:Browse/Old|here]]. [[User:Emesee|Emesee]] 21:13, 1 September 2008 (UTC) == Resource by name and resource by type == How would it be if we removed the "resource by name", put "resource by type" in its place, and then had the departments right below the "Major starting points"? [[User:Emesee|Emesee]] 03:04, 7 September 2008 (UTC) * Darklama's "resource by name" section is a good idea, but it does take up a lot of horizontal space. Rather than "removing" it, perhaps we should just "move" it to a linked subpage - e.g. keep the box, but replace the box content with a hyperlink to the subpage? A second suggestion is that some of the very small boxes could go at the top. --[[User:McCormack|McCormack]] 04:26, 7 September 2008 (UTC) ** I moved the "resource by name" section to a subpage and linked it from the box at the top quite prominently. I haven't done anything about your layout ideas yet, Emesee, but you might like to experiment. --[[User:McCormack|McCormack]] 10:06, 10 September 2008 (UTC) == removing the top box == Let's remove the top box, and put the link to [[Wikiversity:Browse/Old]] somehow in with the headline text ("Major starting points - click here to browse all Portals") on the box below it. [[User:Emesee|Emesee]] 19:41, 8 September 2008 (UTC) * I think the top box is a useful opportunity to orientate users in the use of the page. The wording needs to be brief and appropriate, though. General purposes: tell people what the page is; explain the concept of multiple pathways; explain to people how to get their pages listed. The last of these points is quite important. Although it's actually very easy to get listed (just categorise your pages properly), some editors, including more experienced ones, haven't realised this over the last few months. So a word of help is good as well. --[[User:McCormack|McCormack]] 10:10, 10 September 2008 (UTC) == Browse concept == [[Wikiversity:Browse/Concept]] [[User:Emesee|Emesee]] 22:02, 20 September 2008 (UTC) == Departments == This list is getting quite long. Perhaps we could consider something slightly more heirarchical? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 11:22, 2 November 2008 (UTC) :I think we should merge Schools, Faculties, Departments and Subjects. These categories overlap and should be simplified first, than we could categories from most general to more specific and just list the [[b:Subject:Major Subjects|major subject]] categories like Wikibooks does. --<span style="font: bold 10pt 'courier new', comic, sans, ms;">[[User:Darklama|{{font|color=midnightblue|dark}}]][[Image:Yin yang.svg|12px]][[User_talk:Darklama|{{font|color=green|lama}}]]</span> 19:31, 3 November 2008 (UTC) ::Merging all those might be too much. But perhaps Departments could simply be merged into Schools and Faculties? [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 08:30, 4 November 2008 (UTC) == Overwhelming == I haven't used the "browse" page in quite some time, as I have most of the content I fiddle with watchlisted. I really like the idea of the dynamic lists -- they are powerful and self maintaining. However, they don't work well here. The page is already HUGE, and it will continue to grow (hopefully) as things are added and categorized. Browse should be a good way to quickly and simply identify where you want to go. Now, I'm not suggesting that we take out all of the dynamic lists. But, I know when I first clicked on the page, I was completely lost. It is just so much information, it's hard to take in. The old page, by contrast, while not so simply self-maintaining, was much more compact. I think this actually made browsing itself easier, as you could find your way through the page in a simpler way. The icons were also helpful. Maybe a compromise, where we put out some icons with a few dynamic lists, and perhaps have pointers to other more detailed lists? [[User:Historybuff|Historybuff]] 08:28, 14 February 2009 (UTC) :It was a big mistake to turn the browse page over to automation. Providing a user-friendly guide to Wikiversity content is a job that requires constant community participation. Categorization of pages is just a first step. Every major category should have a user-friendly directory that serves as a guide to the contents of that category. Following the arbitrary convention established long ago at Wikipedia these directories are called [[Wikiversity:Portal|portals]]. Members of the community should devote some of their time to special organizational projects where participants collaborate to organize Wikiversity contents. The major [[Wikiversity:Content development|content development projects]] that support such collaboration are the [[Wikiversity:Schools|Schools]]. --[[User:JWSchmidt|JWSchmidt]] 14:37, 14 February 2009 (UTC) {| style="float: right;" | {{Robelbox|theme=1|title=By Department}} <div style="{{Robelbox/pad}}; font-size:90%" style="height:300px; overflow:auto;"> {{#categorytree:Departments|mode=all|depth=10|hideroot|namespaces=topic category|hideprefix=always}} </div> {{Robelbox/close}} |} ::I am testing this: ::* the ability to limit the size by using a lift at the side, this is 300px, but could be 1000px for departments ::* drill down into sub categories. One sub category that could really be useful would be [[:Category:Department stubs]] for stubs. ::[[User:HenkvD|HenkvD]] 19:58, 27 February 2009 (UTC) == NICE PAGE! == NICE PAGE!!!!!!!! == Extension requested == Hi, 1. I would like to design a course and add content that is MULTIDISCIPLINARY. Can you include that as a category? 2. I am not quite sure of the other aspects of categorisation either. Like school subject (it is not a school subject), or university faculties (could be linguistics, computing or psychology), or the level, or type of resource, because it is being written with a user in mind whose interest lies in solvoing several problems in education. See my intro under [[reflective thinking]], which is just one term asociated with my course planned. --[[User:Genezistan|Genezistan]] 03:45, 13 September 2009 (UTC) :You can actually include your course under ''all'' applicable categories: there is no need to limit yourself! [[user:Jade Knight|The Jade Knight]] ^{[[User talk:Jade Knight|(d'viser)]]} 17:26, 28 September 2009 (UTC) Thanks Adviser [[User:Genezistan|Genezistan]] 03:13, 29 September 2009 (UTC) == Delete school of Accountancy == School of Accounting should be deleted form the list of schools altogether since it doesn't exist.<br /> --[[User:Newmangledmatt|ttam]] ([[User talk:Newmangledmatt|discuss]] • [[Special:Contributions/Newmangledmatt|contribs]]) 00:16, 19 October 2014 (UTC) :It's already gone, but you have to purge the page to see the change. If you want to add the purge feature to your list of 'More', go to Preferences, Gadgets and add the Purge tab. Content like this page that is generated from category lists may not update for some time (hours). If you purge, it is updated immediately. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:43, 19 October 2014 (UTC) 9ec0s78t442nteid8ghgz5ghbdhh7ep 0 203182 2834853 2834016 2026-09-28T12:54:58Z JonAwbrey 29537 /* Truth tables */ resize table 2834853 wikitext text/x-wiki {{font|size=medium|&#9758;}} This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]]. A '''minimal negation operator''' <math>(\nu)</math> is a logical connective which says &ldquo;just one false&rdquo; of its logical arguments.&nbsp; The first four cases are described below. <ol start="0"> <li style="padding:8px"> If the list of arguments is empty, as expressed in the form <math>\nu(),</math> then it cannot be true that exactly one of the arguments is false, so <math>\nu() = \mathrm{false}.</math> </li> <li style="padding:8px"> If <math>p</math> is the only argument then <math>\nu(p)</math> says that <math>p</math> is false, so <math>\nu(p)</math> expresses the logical negation of the proposition <math>p.</math>&nbsp; Written in several different notations, we have the following equivalent expressions. <p style="padding:8px; text-align:center"><math>\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}</math></p> </li> <li style="padding:8px"> If <math>p</math> and <math>q</math> are the only two arguments then <math>\nu(p, q)</math> says that exactly one of <math>p, q</math> is false, so <math>\nu(p, q)</math> says the same thing as <math>p \neq q.</math> Expressing <math>\nu(p, q)</math> in terms of ands <math>(\cdot),</math> ors <math>(\lor),</math> and nots <math>(\tilde{~})</math> gives the following form. <p style="padding:8px; text-align:center"><math>\nu(p, q) ~=~ \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}</math></p> It is permissible to omit the dot <math>(\cdot)</math> in contexts where it is understood, giving the following form. <p style="padding:8px; text-align:center"><math>\nu(p, q) ~=~ \tilde{p}q \lor p\tilde{q}</math></p> The venn diagram for <math>\nu(p, q)</math> is shown in Figure&nbsp;1. {| align="center" cellpadding="8" style="text-align:center" | <p>[[File:Venn Diagram (P,Q).jpg|500px]]</p> <p><math>\text{Figure 1.}~~\nu(p, q)</math></p> |} </li> <li style="padding:8px"> The venn diagram for <math>\nu(p, q, r)</math> is shown in Figure&nbsp;2. {| align="center" cellpadding="8" style="text-align:center" | <p>[[File:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 2.}~~\nu(p, q, r)</math></p> |} The center cell is the region where all three arguments <math>p, q, r</math> hold true, so <math>\nu(p, q, r)</math> holds true in just the three neighboring cells.&nbsp; In other words: <p style="padding:8px; text-align:center"><math>\nu(p, q, r) ~=~ \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}</math></p> </li></ol> ==Initial definition== The <b>minimal negation operator</b> <math>\nu</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k</math> is a <math>k</math>-ary [[boolean function]] defined by the rule that <math>\nu_k (x_1, \ldots, x_k) = 1</math> if and only if exactly one of the arguments <math>x_j</math> is <math>0.</math> In contexts where the initial letter <math>\nu</math> is understood, the minimal negation operators may be indicated by argument lists in parentheses.&nbsp; In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)} = \nu (x, y, z).</math> The first four members of this family of operators are shown below.&nbsp; The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation. {| align="center" style="text-align:center" | [[File:Minimal Negation Operators 0, 1, 2, 3 Arguments 2.0.png|500px]] |} ==Formal definition== To express the general case of <math>\nu_k</math> in terms of familiar operations, it helps to introduce an intermediary concept: '''Definition.'''&nbsp; Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}</math> be defined for each integer <math>j</math> in the interval <math>[1, k]</math> by the following equation: {| align="center" cellpadding="8" | <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math> |} Then <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}</math> is defined by the following equation: {| align="center" cellpadding="8" | <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math> |} If we take the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k</math> to indicate the point <math>x = (x_1, \ldots, x_k)</math> in the space <math>\mathbb{B}^k</math> then the minimal negation <math>\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x</math> in exactly one coordinate.&nbsp; This makes <math>\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}</math> a discrete functional analogue of a point-omitted neighborhood in ordinary real analysis, more exactly, a point-omitted distance-one neighborhood.&nbsp; In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field. The remainder of this discussion proceeds on the algebraic convention that the plus sign <math>(+)</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition mod 2.&nbsp; Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted for logic in such a way that <math>0 = \mathrm{false}</math> and <math>1 = \mathrm{true}.</math>&nbsp; This has the following consequences: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The operation <math>x + y</math> is a function equivalent to the exclusive disjunction of <math>x</math> and <math>y,</math> while its fiber of 1 is the relation of inequality between <math>x</math> and <math>y.</math> |- | valign="top" | <big>&bull;</big> | The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)</math> to its ''parity''. |} The following properties of the minimal negation operators <math>{\nu_k : \mathbb{B}^k \to \mathbb{B}}</math> may be noted: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y</math> and the relation <math>x \ne y.</math> |- | valign="top" | <big>&bull;</big> | In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.</math> |- | valign="top" | <big>&bull;</big> | More generally, the function <math>\nu_k (x_1, \dots, x_k)</math> for <math>k > 2</math> is not identical to the boolean sum <math>\textstyle\sum_{j=1}^k x_j.</math> |- | valign="top" | <big>&bull;</big> | The inclusive disjunctions indicated for the <math>\nu_k</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint. |} ==Truth tables== Table&nbsp;3 is a [[truth table]] for the sixteen boolean functions of type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> whose fibers of 1 are either the boundaries of points in <math>\mathbb{B}^3</math> or the complements of those boundaries. <br> {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%" |+ <math>\text{Table 3. Logical Boundaries and Their Complements}</math> |- style="background:ghostwhite" | <math>\mathcal{L}_1</math> | <math>\mathcal{L}_2</math> | <math>\mathcal{L}_3</math> | <math>\mathcal{L}_4</math> |- style="background:ghostwhite" | &nbsp; | align="right" | <math>p\colon</math> | <math>1~1~1~1~0~0~0~0</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>q\colon</math> | <math>1~1~0~0~1~1~0~0</math> | &nbsp; |- style="background:ghostwhite" | &nbsp; | align="right" | <math>r\colon</math> | <math>1~0~1~0~1~0~1~0</math> | &nbsp; |- | <math>\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}</math> | <math>\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}</math> | <math>\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}</math> | <math>\begin{matrix} \texttt{(}~p~\texttt{,}~q~\texttt{,}~r~\texttt{)} \\[4pt] \texttt{(}~p~\texttt{,}~q~\texttt{,}(r\texttt{))} \\[4pt] \texttt{(}~p~\texttt{,(}q\texttt{),}~r~\texttt{)} \\[4pt] \texttt{(}~p~\texttt{,(}q\texttt{),(}r\texttt{))} \\[4pt] \texttt{((}p\texttt{),}~q~\texttt{,}~r~\texttt{)} \\[4pt] \texttt{((}p\texttt{),}~q~\texttt{,(}r\texttt{))} \\[4pt] \texttt{((}p\texttt{),(}q\texttt{),}~r~\texttt{)} \\[4pt] \texttt{((}p\texttt{),(}q\texttt{),(}r\texttt{))} \end{matrix}</math> |- | <math>\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}</math> | <math>\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}</math> | <math>\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}</math> | <math>\begin{matrix} \texttt{(((}p\texttt{),(}q\texttt{),(}r\texttt{)))} \\[4pt] \texttt{(((}p\texttt{),(}q\texttt{),}~r~\texttt{))} \\[4pt] \texttt{(((}p\texttt{),}~q~\texttt{,(}r\texttt{)))} \\[4pt] \texttt{(((}p\texttt{),}~q~\texttt{,}~r~\texttt{))} \\[4pt] \texttt{((}~p~\texttt{,(}q\texttt{),(}r\texttt{)))} \\[4pt] \texttt{((}~p~\texttt{,(}q\texttt{),}~r~\texttt{))} \\[4pt] \texttt{((}~p~\texttt{,}~q~\texttt{,(}r\texttt{)))} \\[4pt] \texttt{((}~p~\texttt{,}~q~\texttt{,}~r~\texttt{))} \end{matrix}</math> |} <br> ==Charts and graphs== This Section focuses on visual representations of minimal negation operators. A few bits of terminology are useful in describing the pictures, but the formal details are tedious reading, and may be familiar to many readers, so the full definitions of the terms marked in ''italics'' are relegated to a Glossary at the end of the article. Two ways of visualizing the space <math>\mathbb{B}^k</math> of <math>2^k</math> points are the ''hypercube'' picture and the ''venn diagram'' picture. The hypercube picture associates each point of <math>\mathbb{B}^k</math> with a unique point of the <math>k</math>-dimensional hypercube. The venn diagram picture associates each point of <math>\mathbb{B}^k</math> with a unique &ldquo;cell&rdquo; of the venn diagram on <math>k</math> &ldquo;circles&rdquo;. In addition, each point of <math>\mathbb{B}^k</math> is the unique point in the ''[[fiber (mathematics)|fiber]] of truth'' <math>[|s|]</math> of a ''singular proposition'' <math>s : \mathbb{B}^k \to \mathbb{B},</math> and thus it is the unique point where a ''singular conjunction'' of <math>k</math> ''literals'' is <math>1.</math> For example, consider two cases at opposite vertices of the cube: {| align="center" cellpadding="4" width="90%" | valign="top" | <big>&bull;</big> | The point <math>(1, 1, \ldots , 1, 1)</math> with all 1's as coordinates is the point where the conjunction of all posited variables evaluates to <math>1,</math> namely, the point where: |- | &nbsp; | align="center" | <math>x_1 ~ x_2 ~\ldots~ x_{n-1} ~ x_n ~=~ 1.</math> |- | valign="top" | <big>&bull;</big> | The point <math>(0, 0, \ldots , 0, 0)</math> with all 0's as coordinates is the point where the conjunction of all negated variables evaluates to <math>1,</math> namely, the point where: |- | &nbsp; | align="center" | <math>\texttt{(} x_1 \texttt{)(} x_2 \texttt{)} \ldots \texttt{(} x_{n-1} \texttt{)(} x_n \texttt{)} ~=~ 1.</math> |} To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>. Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1</math> on the points adjacent to the point where <math>s</math> is <math>1,</math> and 0 everywhere else on the cube. For example, consider the case where <math>k = 3.</math> Then the minimal negation operation <math>\nu (p, q, r)</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[File:Venn Diagram (P,Q,R).jpg|500px]]</p> <p><math>\text{Figure 4.}~~\texttt{(p, q, r)}</math></p> |} For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram: {| align="center" cellpadding="8" style="text-align:center" | <p>[[File:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p> <p><math>\text{Figure 5.}~~\texttt{((p),(q),(r))}</math></p> |} ==Glossary of basic terms== ; Boolean domain : A ''[[boolean domain]]'' <math>\mathbb{B}</math> is a generic 2-element set, for example, <math>\mathbb{B} = \{ 0, 1 \},</math> whose elements are interpreted as logical values, usually but not invariably with <math>0 = \mathrm{false}</math> and <math>1 = \mathrm{true}.</math> ; Boolean variable : A ''boolean variable'' <math>x</math> is a variable that takes its value from a boolean domain, as <math>x \in \mathbb{B}.</math> ; Proposition : In situations where boolean values are interpreted as logical values, a [[boolean-valued function]] <math>f : X \to \mathbb{B}</math> or a [[boolean function]] <math>g : \mathbb{B}^k \to \mathbb{B}</math> is frequently called a ''proposition''. ; Basis element, Coordinate projection : Given a sequence of <math>k</math> boolean variables, <math>x_1, \ldots, x_k,</math> each variable <math>x_j</math> may be treated either as a ''basis element'' of the space <math>\mathbb{B}^k</math> or as a ''coordinate projection'' <math>x_j : \mathbb{B}^k \to \mathbb{B}.</math> ; Basic proposition : This means that the set of objects <math>\{ x_j : 1 \le j \le k \}</math> is a set of boolean functions <math>\{ x_j : \mathbb{B}^k \to \mathbb{B} \}</math> subject to logical interpretation as a set of ''basic propositions'' that collectively generate the complete set of <math>2^{2^k}</math> propositions over <math>\mathbb{B}^k.</math> ; Literal : A ''literal'' is one of the <math>2k</math> propositions <math>x_1, \ldots, x_k, \texttt{(} x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{)},</math> in other words, either a ''posited'' basic proposition <math>x_j</math> or a ''negated'' basic proposition <math>\texttt{(} x_j \texttt{)},</math> for some <math>j = 1 ~\text{to}~ k.</math> ; Fiber : In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math> : In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers: : The fiber of <math>0</math> under <math>f,</math> defined as <math>f^{-1}(0),</math> is the set of points where the value of <math>f</math> is <math>0.</math> : The fiber of <math>1</math> under <math>f,</math> defined as <math>f^{-1}(1),</math> is the set of points where the value of <math>f</math> is <math>1.</math> ; Fiber of truth : When <math>1</math> is interpreted as the logical value <math>\mathrm{true},</math> then <math>f^{-1}(1)</math> is called the ''fiber of truth'' in the proposition <math>f.</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it. This leads to the definition of the notation <math>[|f|] = f^{-1}(1)</math> for the fiber of truth in the proposition <math>f.</math> ; Singular boolean function : A ''singular boolean function'' <math>s : \mathbb{B}^k \to \mathbb{B}</math> is a boolean function whose fiber of <math>1</math> is a single point of <math>\mathbb{B}^k.</math> ; Singular proposition : In the interpretation where <math>1</math> equals <math>\mathrm{true},</math> a singular boolean function is called a ''singular proposition''. : Singular boolean functions and singular propositions serve as functional or logical representatives of the points in <math>\mathbb{B}^k.</math> ; Singular conjunction : A ''singular conjunction'' in <math>\mathbb{B}^k \to \mathbb{B}</math> is a conjunction of <math>k</math> literals that includes just one conjunct of the pair <math>\{ x_j, ~\nu(x_j) \}</math> for each <math>j = 1 ~\text{to}~ k.</math> : A singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be expressed as a singular conjunction: {| align="center" cellspacing"10" width="90%" | height="36" | <math>s ~=~ e_1 e_2 \ldots e_{k-1} e_k</math>, |- | <math>\begin{array}{llll} \text{where} & e_j & = & x_j \\[6pt] \text{or} & e_j & = & \nu (x_j), \\[6pt] \text{for} & j & = & 1 ~\text{to}~ k. \end{array}</math> |} ==Resources== * [[Logic Syllabus]] * [http://atlas.wolfram.com/ Wolfram Atlas of Simple Programs] ** [http://atlas.wolfram.com/01/01/ Elementary Cellular Automata Rules (ECARs)] ** [http://atlas.wolfram.com/01/01/rulelist.html ECAR Index] ** [http://atlas.wolfram.com/01/01/views/3/TableView.html ECAR Icons] ** [http://atlas.wolfram.com/01/01/views/87/TableView.html ECAR Examples] ** [http://atlas.wolfram.com/01/01/views/172/TableView.html ECAR Formulas] [[Category:Automata theory]] [[Category:Boolean algebra]] [[Category:Boolean functions]] [[Category:Combinatorics]] [[Category:Computation theory]] [[Category:Cybernetics]] [[Category:Differential logic]] [[Category:Discrete mathematics]] [[Category:Equational inference]] [[Category:Graph theory]] [[Category:Logic]] [[Category:Logical graphs]] [[Category:Neural networks]] [[Category:Peirce, Charles Sanders]] [[Category:Propositional calculus]] [[Category:Semiotics]] [[Category:Set theory]] [[Category:Visualization]] [[Category:Minimal Negation Operators]] 80wboz0ig7tixy2j5z8pdstqpjx9viu 1 206506 2834949 2834816 2026-09-29T01:54:20Z Nv0011 3111788 /* Associate editor application of Neil Vedwan */ 2834949 wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2016]]<br>[[/Archive 2017]]<br>[[/Archive 2018]]<br>[[/Archive 2019]]<br>[[/Archive 2020]]<br>[[/Archive 2022]]<br>[[/Archive 2023]] }} }} </noinclude> ==Associate editor application of OGUNJIMI OLUWOLE TEMIDAYO== {{WikiJournal editor application submitted | position =Associate editor | name =OGUNJIMI OLUWOLE TEMIDAYO | qualifications =M.Sc. Geography and Planning | link =Linkedin.com/in/oluwole-ogunjimi | areas_of_expertise =Enivironmental Management and Data Analysis | professional_experience =An administrator with over five years of working experience in both healthcare and education sectors. I have a Master’s degree from the University of Lagos and a Bachelor's degree from the University of Ibadan, I have developed significant research skills, especially in analyzing the impacts of human activities on water quality and understanding the socio-economic effects of environmental changes. With hands-on experience in geospatial mapping, data analytics, and programming (Python), he combines his passion for environmental conservation with a commitment to translating research findings into actionable insights. | publishing_experience =Environmental and Socio-Economic Impact of Artisanal Mining in Oke Ogun Region, 2016. https://drive.google.com/file/d/1l9WTh-ZhUpKbRN_TP28oP-mvlaiWGf0S/view?usp=drive_link Impact of Human Activities on Water Quality of Kudeti River, Ibadan, Oyo State, Nigeria, 2016. https://drive.google.com/file/d/1N00Eo1RitC5dKoX38G1H8r7MJsKdD3Aw/view?usp=drive_link Impact of Rural Transportation on The Distribution of Agricultural Produce in Obafemi Owode Local Government Area, Ogun State, Nigeria, 2023. https://drive.google.com/file/d/1YMYpcdC1JqnxQO-fq5eM3eitbHpQH-HB/view?usp=drive_link | open_experience =I am an active Wikipedia editor with over 300 edits. Significant editing history on English Wikipedia under username "Kamoranesi90". | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 12:35, 18 October 2024 (UTC) }} * {{re|Kamoranesi90}} Please adjust your security settings for the 3 Google Drive links. I am unable to view them. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:32, 25 October 2024 (UTC) *:Thank you @[[User:OhanaUnited|OhanaUnited]]. I have adjusted the security settings. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 14:49, 25 October 2024 (UTC) *::Thank you. Do you have any publication examples which are peer-reviewed? For example, these could be journal articles or book chapters. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:12, 25 October 2024 (UTC) *:::No, I don't but I am working on publishing my research works. Thank you [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 09:19, 26 October 2024 (UTC) *::::I am open to '''support''' your associate editor application with mentorship from board members to support your growth in gaining experience around the publishing area. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:52, 14 November 2024 (UTC) *:::::Wow. That's great. I am open to learning from your wealth of knowledge and experience. I appreciate your kind gesture. *:::::Thank you. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 18:59, 14 November 2024 (UTC) *::::::Please wait for other editorial board members to review and comment on your application. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:44, 18 November 2024 (UTC) *:::::::Thank you. [[User:Kamoranesi90|Kamoranesi90]] ([[User talk:Kamoranesi90|discuss]] • [[Special:Contributions/Kamoranesi90|contribs]]) 02:59, 22 November 2024 (UTC) *If [[User:OhanaUnited]] is happy to mentor, I think you would be a valuable addition to the team, so I would be willing to support. Associate editor positions don't need to be as experienced as board members, just willing to learn and able to work well with others, especially if someone is able to take the lead if calleging scenarios arise (e.g. confilcting peer reviews). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 9 January 2025 (UTC) *'''Support application''' Though I lack experience in such matters, I have recently learned enough to recognize this to be an easy decision. Regarding mentoring, I think I could use some mentoring myself. Whenever appropriate, please CC me on mentor-mentee communications. In particular, I am having trouble dealing with the large number of referee invites required to recruit volunteers. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:50, 11 January 2025 (UTC) {{re|Kamoranesi90}} My apologies for the delay in getting back to you. '''Result: Accepted into the editorial board as associate editor.''' : [[WikiJournal User Group/Editorial guidelines#Adding editorial board members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) {{clickable button 2|Onboarding email template|url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member}} # Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # Add their name and start data to the [d:{{WJQboard|default=Q75674277}} relevant editorial board] on wikidata # Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # Welcome them at the {{#if:|wjm|WJM}}board mailing list so that they are informed # Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:12, 2 April 2025 (UTC) {{ping|Kamoranesi90}}, can you confirm if you're still interested in joining? I have emailed you through the wiki's email function a while back but did not receive a response from you. I also noticed that [https://guc.toolforge.org/?by=date&user=Kamoranesi90 you haven't edited in any WMF projects since November 2024] so I don't know if you still monitor your account for notifications. If I don't hear from you by the end of this month, I will archive this application without prejudice (i.e. you are more than welcome to continue the process to join as an associate editor when you replied back to this message). [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:43, 2 January 2026 (UTC) ==Editorial board application of Max Loomes== {{WikiJournal editor application submitted | position =Editorial board | name =Max Loomes | qualifications =Bachelor of psychology (Hons I)(USYD), Master of Clinical Psychology (UTS), PhD Candidate (UTS) | link =https://profiles.uts.edu.au/student_Max.W.Loomes | areas_of_expertise =Mental health, psychiatry, public health | professional_experience =Research (9 years, ~4 institutions, various areas both qualitative and quantitative, field work [ecology/education psychology], preclinical trials, project managing, clinical practice (trauma, schizophrenia), teaching (~1 year at 2 institutions for psychology undergraduate and post-graduate, some masters). | publishing_experience =Published ~12 articles/chapters, reviewed for three different journals | open_experience =Have updated various wikipedia pages and big believer in open access and dismantling the highly capitalistic structures that govern journal publishing | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Maxwloomes|Maxwloomes]] ([[User talk:Maxwloomes|discuss]] • [[Special:Contributions/Maxwloomes|contribs]]) 06:34, 13 May 2025 (UTC) }} :<nowiki>Great to see this application! I support it; have known Max for several years and he is passionate about many of the values behind WikiJournal ~~~</nowiki> [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 20:46, 4 February 2026 (UTC) --[[User:Maxwloomes|Maxwloomes]] ([[User talk:Maxwloomes|discuss]] • [[Special:Contributions/Maxwloomes|contribs]]) 06:34, 13 May 2025 (UTC) :* '''Support'''. Applicant has experience in research, publishing and open access (both Wikipedia and journal). [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 00:23, 27 May 2025 (UTC) ==Editorial board application of Alex O. Holcombe== {{WikiJournal editor application submitted | position =Editorial board | name =Alex O. Holcombe | qualifications =PhD in psychology | link =https://profiles.sydney.edu.au/alex.holcombe | areas_of_expertise =Psychology, neuroscience | professional_experience =Full-time academic position for 20 years | publishing_experience =Advisory board or editorial board member of journals and initiatives including PLOS ONE, Collabra:Psychology, Replication Research, Meta-ROR, Meta-psychology... | open_experience =Have been an Associate Editor of WikiJournalOfScience for the last couple years, handled three articles to publication, and have attended most, I think, of the monthly meetings in the last two years | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[User:Aoholcombe|Aoholcombe]] ([[User talk:Aoholcombe|discuss]] • [[Special:Contributions/Aoholcombe|contribs]]) 20:56, 4 February 2026 (UTC) }} : I '''strongly support''' promoting Alex to be a full editorial board member. He has the handling editor for a number of submissions, e.g. [[WikiJournal of Science/Hypericum sechmenii|1]], [[WikiJournal of Science/The Himalayan fossil hoax|2]] and [[WikiJournal of Science/Popular Pet Reptile, the Leopard Gecko (Eublepharis macularius), Spontaneously Uses Running Wheel– Is It Locomotion Play?|3]]). He also gave insightful comments towards the operations of the WikiJournal of Science editorial board and the broader WikiJournal publishing umbrella. We need the expertise like Alex to further strengthen the board's expertise. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:03, 5 February 2026 (UTC) ==Associate editor application of Neil Vedwan== {{WikiJournal editor application submitted | position =Associate editor | name =Neil Vedwan | qualifications = | link = | areas_of_expertise =Type theory, Formal Methods, Computational Complexity, Computability, Category Theory, Set Theory | professional_experience = Independent Researcher (in collaboration with faculty @ Montclair State University) | publishing_experience =Reviewer at zbMath Open, In the process of getting 2 papers published | open_experience =Contributor to CSLib (a library of Lean 4), Wikipedia contributor | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Science. [[Special:Contributions/~2026-52029-76|~2026-52029-76]] ([[User talk:~2026-52029-76|talk]]) 03:53, 28 September 2026 (UTC) }} :It looks like you're not logged in. Please log in and attach your signature to this application. Furthermore, I am unable to find a research by this name in Montclair State University's website with your area of expertise. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 06:52, 28 September 2026 (UTC) e8wb0mdqvn5hxlqjenleo2wv2dtwed6 2 219126 2834873 2815178 2026-09-28T19:16:06Z ThaniosAkro 2805358 /* Implementation */ 2834873 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ==Complex Cube Root== See [https://en.wikiversity.org/wiki/Cubic_function#Vieta's_substitution Vieta's Substitution.] The depressed cubic function is : <math>f(t) = t^3 + At + B = 0.</math> Let <math>A = -3C</math> and let <math>t = w + \frac{C}{w} = \frac{w^2 + C}{w}.</math> Substitute for <math>A, t</math> in the depressed function and result is: <math>f(w) = w^6 + Bw^3 + C^3</math> or <math>f(W) = W^2 + BW + C^3</math> where <math>W = w^3</math> and <math>w = \sqrt[3]{W}</math>. From the quadratic formula: <math>W = \frac{-B \pm \sqrt{B^2 - 4C^3}}{2} = \frac{-B \pm \sqrt{4C^3 - B^2}\sqrt{-1} }{2}</math> However, <math>W = a + bi.</math> Therefore: <math>a = \frac{-B}{2}</math> and <math>b = \frac{ \sqrt{4C^3 - B^2} }{2}.</math> <math>B = -2a</math> and <math>2b = \sqrt{4C^3 - B^2} </math> <math>4b^2 = 4C^3 - B^2 </math> <math>4C^3 = 4b^2 + B^2 </math> <math>C^3 = b^2 + \frac{B^2}{4} </math> <math>C = \sqrt[3]{C^3} </math> and <math>A = -3C. </math> <math>A,B </math> of <math>f(t)</math> have been defined with <math>C</math> positive and <math>A</math> negative. Because <math>W</math> has three complex roots, <math>f(t)</math> must have three real roots. Calculate one of the roots of <math>f(t).</math> <math>t = \frac{w^2 + C}{w}.</math> Therefore <math>wt = w^2 + C</math> and <math>w^2 - tw + C = 0.</math> From the quadratic formula : <math>w = \frac{t \pm \sqrt{t^2 - 4C}}{2} = \frac{t \pm \sqrt{4C - t^2}\sqrt{-1} }{2}</math> <math>w = p + qi</math> Therefore: <math>p = \frac{t}{2}</math> and <math>Q = \frac{4C - t^2 }{4} = C - \frac{t^2}{4}</math> where <math>Q = q^2.</math> <math>q = \sqrt{Q}</math> but sign of this calculation of <math>q</math> is ambiguous. Let <math>P = p^2</math> <math>q = \frac{b}{(3P - Q)}</math> and <math>(p + qi)^3 = a + bi.</math> ===An Example=== Calculate cube roots of complex number <math>W = -39582 + 3799i.</math> {{RoundBoxTop|theme=2}} [[File:1215depressed cubic01.png|thumb|400px|'''Graph of <math>f(t)</math> shown as graph of <math>f(x)</math> and showing three values of <math>t: t_1, t_2, t_3</math>.''' </br> <math>Y</math> axis compressed for clarity. </br> <math>A,B = -3495.0, 79164.0</math> </br> <math>t_1 = -68.229473419497441512295943903670298\dots</math> </br> <math>t_2 = 32.229473419497441512295943903670298\dots</math> </br> <math>\ t_3 = 36</math> ]] <syntaxhighlight lang=python> # python code: import decimal decimal.getcontext().prec = 22 ab = -39582,3799 a,b = [ dD(v) for v in ab ] B = -2*a C = (b**2 + B*B/4) ** (dD(1)/3) A = -3*C a,b,A,B,C = [ dD(str(float(v))) for v in (a,b,A,B,C) ] print ( 'a,b = {}, {}'.format(a,b) ) print ( 'A,B,C = {}, {}, {}'.format(A,B,C) ) </syntaxhighlight> <syntaxhighlight> a,b = -39582.0, 3799.0 A,B,C = -3495.0, 79164.0, 1165.0 </syntaxhighlight> Calculate roots of cubic function: <math>y = f (t) </math><math> = t^3 </math><math> - 3495 t </math><math> + 79164 .</math> Three roots are: <math>t_1,\ t_2,\ t_3 = -68.22947341949744\dots,\ 32.22947341949744\dots,\ 36.0</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code: t1, t2, t3 = ('-68.229473419497441512295943903670298', '32.229473419497441512295943903670298', 36 ) values_of_t = [ dD(v) for v in (t1,t2,t3) ] for t in values_of_t : p = t/2 ; P = p**2 Q = C - t*t/4 ; q = b/(3*P - Q) # Check results: print () sx = 't' ; print (sx,'=', eval(sx)) print ( 'p,q = {}, {}'.format(p,q) ) ab = [ p*P - 3*p*Q, 3*P*q - q*Q ] a_,b_ = [ float(v) for v in ab ] sx = 'a_,b_' ; print (sx,'=', eval(sx)) </syntaxhighlight> <syntaxhighlight> t = -68.229473419497441512295943903670298 p,q = -34.11473670974872075615, 1.088457268119895641747 a_,b_ = (-39582.0, 3799.0) t = 32.229473419497441512295943903670298 p,q = 16.11473670974872075615, -30.08845726811989564176 a_,b_ = (-39582.0, 3799.0) t = 36 p,q = 18, 29 a_,b_ = (-39582.0, 3799.0) </syntaxhighlight> Three cube roots of <math>W = -39582 + 3799i</math> are: <math>w_1 = -34.11473670974872075615 + 1.088457268119895641747i</math> <math>w_2 = 16.11473670974872075615 - 30.08845726811989564176i</math> <math>w_3 = 18 + 29i</math> ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> s49j76f66cdfrh8zjsxckym2e2dqj3q 2834874 2834873 2026-09-28T19:16:53Z ThaniosAkro 2805358 /* Complex Cube Root */ 2834874 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 2gb6q898rlyo1rjzq104du19bv5l3e1 2834876 2834874 2026-09-28T19:21:19Z ThaniosAkro 2805358 /* Table */ 2834876 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 01mjnq6buqnyrugbxy3nj89x21y247p 2834883 2834876 2026-09-28T19:44:52Z ThaniosAkro 2805358 /* First result */ 2834883 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> <syntaxhighlight> # Success: </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> chc8twyonrmusl634y3n85a7c91ws8j 2834886 2834883 2026-09-28T19:49:47Z ThaniosAkro 2805358 /* First result */ 2834886 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> <syntaxhighlight> # Success: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that 170, i73 are negative. </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> jorgkj5zik710oovuee9kgbsd954v7y 2834887 2834886 2026-09-28T19:52:46Z ThaniosAkro 2805358 /* First result */ 2834887 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that 170, i73 are negative. </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 4p3e2q6cr22mob9ffpo5ssbp9iq6hja 2834892 2834887 2026-09-28T20:05:00Z ThaniosAkro 2805358 /* First result */ 2834892 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that 170, i73 are negative. </syntaxhighlight> =====Check all conditions===== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 3red640a0yqqmc3oytpmuuwveeyuh0q 2834897 2834892 2026-09-28T20:12:24Z ThaniosAkro 2805358 /* Check all conditions */ 2834897 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that 170, i73 are negative. </syntaxhighlight> =====Check all conditions===== <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; 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We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> kci806yg6rdifv9fsfjaqwxm33tdkh7 2834898 2834897 2026-09-28T20:15:05Z ThaniosAkro 2805358 /* Success */ 2834898 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; 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We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> p7lqoddfoldqtbvhegu1egbd46sw06t 2834900 2834898 2026-09-28T20:29:25Z ThaniosAkro 2805358 /* Check all conditions */ 2834900 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Produce results: <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> ald1bhx4dy3t0px3cdf9b5wt648zv5g 2834902 2834900 2026-09-28T20:33:03Z ThaniosAkro 2805358 /* First result */ 2834902 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' if '=' in input_str : if '==' in input_str : splitter = '==' else : splitter = '=' str1,str2 = input_str.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; 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We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> kgywjtj14ydd6rapfb5ww76jwsab3kr 2834905 2834902 2026-09-28T20:49:41Z ThaniosAkro 2805358 /* Check all conditions */ 2834905 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str1_ = ' '.join(input_str.split()) if '=' in str1_ : if '==' in str1_ : splitter = '==' else : splitter = '=' str1,str2 = str1_.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; 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We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> odzyyiv2jlr3a8r6j6f1m5857685b1s 2834909 2834905 2026-09-28T21:12:33Z ThaniosAkro 2805358 /* Calculate R */ 2834909 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str1_ = ' '.join(input_str.split()) if '=' in str1_ : if '==' in str1_ : splitter = '==' else : splitter = '=' str1,str2 = str1_.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> Produce matrix corrected: <syntaxhighlight lang=python> print ('\nCreating new matrix with corrections:') matrix1 = [ list(v) for v in matrix ] # Deep copy. # e70, i70 are at position 60 in each row of matrix1. # e73, i73 are at position 63 in each row of matrix1. for row in matrix : for p in (60,63) : row[p] *= -1 </syntaxhighlight> Circuit parameters: <syntaxhighlight> values_of_circuit_current = ('i40 + i72 + i35', 'i14 + i71 + i69') values_of_circuit_voltage = ( 'e72 + e46 + e47 + e48 + e70 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e73 + e61 + e62 + e63 + e71', 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69', 'e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69', 'e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69', 'e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69', 'e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69', 'e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14', 'e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14', 'e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14', 'e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14', 'e72 + e31 - e50 + e37 + e38 + e39 + e73 + e61 - e28 + e57' + ( '- e22 + e53 - e16 + e70 + e10 + e11 + e12 + e13 + e14') ) </syntaxhighlight> Calculate values of R: <syntaxhighlight lang=python> Values_of_R = [] for I in range(10, 74): new_line = [ (0) ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = (1) i = (I)/10 new_line[-1] = -i matrix2[-1] = new_line values_of_i = solveMbyN (matrix2) values_of_e = [ i*r for i,r in zip ( values_of_i, values_of_R ) ] ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = values_of_e for Istr in values_of_circuit_current : I = eval(Istr) for Estr in values_of_circuit_voltage : E = eval(Estr) ; R = E/I ; Values_of_R += [R] print ('Number of tests =', len(Values_of_R)) tmp = sorted(Values_of_R) minR = tmp[0] ; maxR = tmp[-1] s1 = 'minR' ; print(s1,'=',eval(s1)) s1 = 'maxR' ; print(s1,'=',eval(s1)) sum = sum_zero((minR, -maxR), 1e-14) s1 = 'sum' ; print(s1,'=',eval(s1)) </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 6bygqbsmom9bfffjv4l0k6ge9swmved 2834911 2834909 2026-09-28T21:17:56Z ThaniosAkro 2805358 /* Calculate R */ 2834911 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str1_ = ' '.join(input_str.split()) if '=' in str1_ : if '==' in str1_ : splitter = '==' else : splitter = '=' str1,str2 = str1_.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(input_str.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> Produce matrix corrected: <syntaxhighlight lang=python> print ('\nCreating new matrix with corrections:') matrix1 = [ list(v) for v in matrix ] # Deep copy. # e70, i70 are at position 60 in each row of matrix1. # e73, i73 are at position 63 in each row of matrix1. for row in matrix1 : for p in (60,63) : row[p] *= -1 </syntaxhighlight> Circuit parameters: <syntaxhighlight> values_of_circuit_current = ('i40 + i72 + i35', 'i14 + i71 + i69') values_of_circuit_voltage = ( 'e72 + e46 + e47 + e48 + e70 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e73 + e61 + e62 + e63 + e71', 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69', 'e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69', 'e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69', 'e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69', 'e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69', 'e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14', 'e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14', 'e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14', 'e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14', 'e72 + e31 - e50 + e37 + e38 + e39 + e73 + e61 - e28 + e57' + ( '- e22 + e53 - e16 + e70 + e10 + e11 + e12 + e13 + e14') ) </syntaxhighlight> Calculate values of R: <syntaxhighlight lang=python> Values_of_R = [] number_of_columns = 65 for I in range(10, 74): new_line = [ 0 ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = (1) i = (I)/10 new_line[-1] = -i matrix1[-1] = new_line values_of_i = solveMbyN (matrix1) values_of_e = [ i*r for i,r in zip ( values_of_i, values_of_R ) ] ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = values_of_e for Istr in values_of_circuit_current : I = eval(Istr) for Estr in values_of_circuit_voltage : E = eval(Estr) ; R = E/I ; Values_of_R += [R] print ('Number of tests =', len(Values_of_R)) tmp = sorted(Values_of_R) minR = tmp[0] ; maxR = tmp[-1] s1 = 'minR' ; print(s1,'=',eval(s1)) s1 = 'maxR' ; print(s1,'=',eval(s1)) sum = sum_zero((minR, -maxR), 1e-14) s1 = 'sum' ; print(s1,'=',eval(s1)) </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 86t7dug8s58brklqqibpvmaid8bvs21 2834915 2834911 2026-09-28T21:35:44Z ThaniosAkro 2805358 /* Check all conditions */ 2834915 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str1_ = ' '.join(input_str.split()) if '=' in str1_ : if '==' in str1_ : splitter = '==' else : splitter = '=' str1,str2 = str1_.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(str1_.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> Produce matrix corrected: <syntaxhighlight lang=python> print ('\nCreating new matrix with corrections:') matrix1 = [ list(v) for v in matrix ] # Deep copy. # e70, i70 are at position 60 in each row of matrix1. # e73, i73 are at position 63 in each row of matrix1. for row in matrix1 : for p in (60,63) : row[p] *= -1 </syntaxhighlight> Circuit parameters: <syntaxhighlight> values_of_circuit_current = ('i40 + i72 + i35', 'i14 + i71 + i69') values_of_circuit_voltage = ( 'e72 + e46 + e47 + e48 + e70 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e73 + e61 + e62 + e63 + e71', 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69', 'e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69', 'e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69', 'e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69', 'e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69', 'e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14', 'e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14', 'e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14', 'e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14', 'e72 + e31 - e50 + e37 + e38 + e39 + e73 + e61 - e28 + e57' + ( '- e22 + e53 - e16 + e70 + e10 + e11 + e12 + e13 + e14') ) </syntaxhighlight> Calculate values of R: <syntaxhighlight lang=python> Values_of_R = [] number_of_columns = 65 for I in range(10, 74): new_line = [ 0 ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = (1) i = (I)/10 new_line[-1] = -i matrix1[-1] = new_line values_of_i = solveMbyN (matrix1) values_of_e = [ i*r for i,r in zip ( values_of_i, values_of_R ) ] ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = values_of_e for Istr in values_of_circuit_current : I = eval(Istr) for Estr in values_of_circuit_voltage : E = eval(Estr) ; R = E/I ; Values_of_R += [R] print ('Number of tests =', len(Values_of_R)) tmp = sorted(Values_of_R) minR = tmp[0] ; maxR = tmp[-1] s1 = 'minR' ; print(s1,'=',eval(s1)) s1 = 'maxR' ; print(s1,'=',eval(s1)) sum = sum_zero((minR, -maxR), 1e-14) s1 = 'sum' ; print(s1,'=',eval(s1)) </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 50d4ah4bs5y9vcpm1459smkyflpqmjr 2834917 2834915 2026-09-28T21:56:42Z ThaniosAkro 2805358 /* Calculate R */ 2834917 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str1_ = ' '.join(input_str.split()) if '=' in str1_ : if '==' in str1_ : splitter = '==' else : splitter = '=' str1,str2 = str1_.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(str1_.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') sum, condition = (6.821210263296962e-13, '+ e56 + e28 - e33 - e61') sum, condition = (-1.1368683772161603e-12, '+ e58 + e18 - e23 - e63') sum, condition = (1.8189894035458565e-12, '+ e64 + e14 - e19 - e69') sum, condition = (1.9326762412674725e-12, '+ e44 + e70 - e15') sum, condition = (2.2737367544323206e-12, '+ e64 + e14 - e71') sum, condition = (2.5011104298755527e-12, '+ e40 + e30 - e72') sum, condition = (1.7053025658242404e-12, '+ e49 + e70 - e10') sum, condition = (3.183231456205249e-12, '+ e35 + e45 - e72') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> Produce matrix corrected: <syntaxhighlight lang=python> print ('\nCreating new matrix with corrections:') matrix1 = [ list(v) for v in matrix ] # Deep copy. # e70, i70 are at position 60 in each row of matrix1. # e73, i73 are at position 63 in each row of matrix1. for row in matrix1 : for p in (60,63) : row[p] *= -1 </syntaxhighlight> Circuit parameters: <syntaxhighlight> values_of_circuit_current = ('i40 + i72 + i35', 'i14 + i71 + i69') values_of_circuit_voltage = ( 'e72 + e46 + e47 + e48 + e70 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e73 + e61 + e62 + e63 + e71', 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69', 'e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69', 'e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69', 'e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69', 'e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69', 'e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14', 'e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14', 'e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14', 'e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14', 'e72 + e31 - e50 + e37 + e38 + e39 + e73 + e61 - e28 + e57' + ( '- e22 + e53 - e16 + e70 + e10 + e11 + e12 + e13 + e14') ) </syntaxhighlight> Calculate 1664 values of R: <syntaxhighlight lang=python> Values_of_R = [] number_of_columns = 65 for I in range(10, 74): new_line = [ 0 ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = (1) i = (I)/10 new_line[-1] = -i matrix1[-1] = new_line values_of_i = solveMbyN (matrix1) values_of_e = [ i*r for i,r in zip ( values_of_i, values_of_R ) ] ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = values_of_e for Istr in values_of_circuit_current : I = eval(Istr) for Estr in values_of_circuit_voltage : E = eval(Estr) ; R = E/I ; Values_of_R += [R] print ('Number of tests =', len(Values_of_R)) tmp = sorted(Values_of_R) minR = tmp[0] ; maxR = tmp[-1] s1 = 'minR' ; print(s1,'=',eval(s1)) s1 = 'maxR' ; print(s1,'=',eval(s1)) sum = sum_zero((minR, -maxR), 1e-14) s1 = 'sum' ; print(s1,'=',eval(s1)) </syntaxhighlight> <syntaxhighlight> Number of tests = 1664 minR = 723.9285759884408 maxR = 723.9285759884439 sum = 0.0 </syntaxhighlight> This code produced 1664 values of R from minR through maxR, both included. With tolerance of <math>1e-14</math> all values of resistance R = 723.92857598844. <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; 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We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> b69mnsjnjje2mjj99qnd7jpx9un35wc 2834977 2834917 2026-09-29T07:31:10Z ThaniosAkro 2805358 /* Check all conditions */ 2834977 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str0 = ' '.join(input_str.split()) if '=' in str0 : if '==' in str0 : splitter = '==' else : splitter = '=' str1,str2 = str0.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(str0.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> Produce matrix corrected: <syntaxhighlight lang=python> print ('\nCreating new matrix with corrections:') matrix1 = [ list(v) for v in matrix ] # Deep copy. # e70, i70 are at position 60 in each row of matrix1. # e73, i73 are at position 63 in each row of matrix1. for row in matrix1 : for p in (60,63) : row[p] *= -1 </syntaxhighlight> Circuit parameters: <syntaxhighlight> values_of_circuit_current = ('i40 + i72 + i35', 'i14 + i71 + i69') values_of_circuit_voltage = ( 'e72 + e46 + e47 + e48 + e70 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e73 + e61 + e62 + e63 + e71', 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69', 'e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69', 'e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69', 'e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69', 'e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69', 'e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14', 'e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14', 'e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14', 'e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14', 'e72 + e31 - e50 + e37 + e38 + e39 + e73 + e61 - e28 + e57' + ( '- e22 + e53 - e16 + e70 + e10 + e11 + e12 + e13 + e14') ) </syntaxhighlight> Calculate 1664 values of R: <syntaxhighlight lang=python> Values_of_R = [] number_of_columns = 65 for I in range(10, 74): new_line = [ 0 ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = (1) i = (I)/10 new_line[-1] = -i matrix1[-1] = new_line values_of_i = solveMbyN (matrix1) values_of_e = [ i*r for i,r in zip ( values_of_i, values_of_R ) ] ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = values_of_e for Istr in values_of_circuit_current : I = eval(Istr) for Estr in values_of_circuit_voltage : E = eval(Estr) ; R = E/I ; Values_of_R += [R] print ('Number of tests =', len(Values_of_R)) tmp = sorted(Values_of_R) minR = tmp[0] ; maxR = tmp[-1] s1 = 'minR' ; print(s1,'=',eval(s1)) s1 = 'maxR' ; print(s1,'=',eval(s1)) sum = sum_zero((minR, -maxR), 1e-14) s1 = 'sum' ; print(s1,'=',eval(s1)) </syntaxhighlight> <syntaxhighlight> Number of tests = 1664 minR = 723.9285759884408 maxR = 723.9285759884439 sum = 0.0 </syntaxhighlight> This code produced 1664 values of R from minR through maxR, both included. With tolerance of <math>1e-14</math> all values of resistance R = 723.92857598844. <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;"> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> ====Welcomee==== {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFF800; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> 3b0id6rj5vhlnnnkossmak94njmcemf 2834987 2834977 2026-09-29T07:39:18Z ThaniosAkro 2805358 /* Calculate R */ 2834987 wikitext text/x-wiki ==Table== {| class="wikitable" |- | From <math>(3):</math> || <math>p^3</math>|| || <math>+Ap</math> || <math>+B</math>|| <math>= 0</math> || <math>\dots (5)</math> |- | From <math>(4):</math> || || <math>Cp^2</math>|| <math></math> || <math>+D</math>|| <math>= 0</math> || <math>\dots (6)</math> |- | <math>(5)*D</math> || <math>Dp^3</math>|| || <math>+DAp</math> || <math>+DB</math>|| <math>= 0</math> || <math>\dots (7)</math> |- | <math>(6)*B</math> || || <math>BCp^2</math>|| <math></math> || <math>+BD</math>|| <math>= 0</math> || <math>\dots (8)</math> |- | <math>(7)-(8)</math> || <math>Dp^3</math>||<math>-BCp^2</math> || <math>+DAp</math> || <math></math>|| <math>= 0</math> || <math>\dots (9)</math> |- | Simplify <math>(9)</math> || <math></math>||<math>Dp^2</math> || <math>-BCp</math> || <math>+DA</math>|| <math>= 0</math> || <math>\dots (10)</math> |- | <math>(6)*A</math> || || <math>ACp^2</math>|| <math></math> || <math>+AD</math>|| <math>= 0</math> || <math>\dots (11)</math> |- | <math>(10)-(11)</math> || <math></math>||<math>Dp^2-ACp^2</math> || <math>-BCp</math> || <math></math>|| <math>= 0</math> || <math>\dots (12)</math> |- | Simplify <math>(12)</math> || <math></math>|| || <math>Dp-ACp</math> || <math>-BC</math> || <math>= 0</math> || <math>\dots (13)</math> |} ====First result==== <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> Calculate values_of_i : <syntaxhighlight lang=python> print ('\nWhen ints become huge:') values_of_i = solveMbyN (matrix) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> When ints become huge: solveMbyN (input_, level = -41) : <class 'OverflowError'> OverflowError('int too large to convert to float') Length of input = 23 input_[0] = Error in this row. input_[1] = Error in this row. input_[2] = Error in this row. input_[3] = [-7.048390979785929e+176, 1.1003906499643917e+176, 1.5070856217200028e+174, 1.955699032422673e+173, ...... 0.0] input_[4] = [510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[5] = [0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[6] = [0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, 220.0, ...... 0.0] input_[7] = Error in this row. values_of_i = None </syntaxhighlight> <syntaxhighlight lang=python> print ('\nConvert input to floats:') matrix1 = [ list(v) for v in matrix ] matrix1[0][0] = float(matrix1[0][0]) values_of_i = solveMbyN (matrix1) sx = 'values_of_i' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> Convert input to floats: solveMbyN (input_, level = -40) : <class 'KeyError'> KeyError('Internal error. input[1][0] = nan, should be 0.') Length of input = 24 input_[0] = [-1.7360455505275518e+174, 1.4096781959571856e+174, -2.2007812999287836e+173, -3.014171243439967e+171, ...... 0.0] input_[1] = [-3.7974175914221006e+162, 3.309163202261842e+161, 2.4114748810711523e+162, -4.426635506031073e+161, ...... 0.0] input_[2] = [-2.4975968400753463e+139, 5.704494025143264e+138, -2.7146889822895123e+138, 1.4057733058789272e+139, ...... 0.0] input_[3] = [7.667237322891148e+182, -1.7692205437935546e+182, -2.341735229706114e+180, 4.1615580570562045e+181, ...... -0.0] input_[4] = [-500.0, 0.0, 0.0, 0.0, 0.0, 1240.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, 0.0, 0.0, 0.0, 690.0, -320.0, ...... 0.0] input_[5] = [0.0, 510.0, 0.0, 0.0, 0.0, 320.0, -1150.0, 270.0, 0.0, 0.0, 320.0, -590.0, 270.0, 0.0, 0.0, 320.0, -590.0, ...... 0.0] input_[6] = [0.0, 0.0, 520.0, 0.0, 0.0, 0.0, 270.0, -1060.0, 220.0, 0.0, 0.0, 270.0, -490.0, 220.0, 0.0, 0.0, 270.0, ...... 0.0] input_[7] = [0.0, 0.0, 0.0, 530.0, 0.0, 0.0, 0.0, 220.0, -970.0, 170.0, 0.0, 0.0, 220.0, -390.0, 170.0, 0.0, 0.0, ...... 0.0] values_of_i = None </syntaxhighlight> =====Success===== <syntaxhighlight lang=python> print ('\nSet global variable enable_reduceRow:') enable_reduceRow = 1 values_of_i = solveMbyN (matrix1) </syntaxhighlight> Success: <syntaxhighlight> Set global variable enable_reduceRow: i10 = 1.0000000000000049 ; i11 = 1.643236374952256 ; i12 = 2.316468657232291 ; i13 = 3.0507456069545116 i14 = 3.908085797412319 ; i15 = 0.5970301043016677 ; i16 = 0.8274774533758196; i17 = 1.225283946841065 i18 = 1.5617922630400025 ; i19 = 0.369273540611037 ; i20 = 0.3858385105885722; i21 = 0.6270001063825295 i22 = 0.7040607630623028 ; i23 = 0.6129764173251202 ; i24 = 0.298576896960036 ; i25 = 0.44960726241462035 i26 = 0.7812212615819403 ; i27 = 0.7165761045550649 ; i28 = 0.5186812635664733; i29 = 0.266770119423805 i30 = 0.5896680016803755 ; i31 = 1.475274068751303 ; i32 = 0.9632057010422588; i33 = 0.5801910619990465 i34 = 0.30799558703953794; i35 = 2.4693659910366916 ; i36 = 1.583773707944257 ; i37 = 1.0123878002551232 i38 = 0.6135963601029517 ; i39 = 0.3175434839291887 ; i40 = 2.7147355593758107; i41 = 2.125067557695435 i42 = 1.6754602952808149 ; i43 = 1.2896217846922426 ; i44 = 0.692591680390575 ; i45 = 0.8855922830924343 i46 = 1.7538676385971101 ; i47 = 1.4222536394297902 ; i48 = 1.181092043635833 ; i49 = 0.6432363749522512 i50 = 0.5713859076891339 ; i51 = 1.0834542753981782 ; i52 = 1.1480994324250535; i53 = 1.0710387757452802 i54 = 0.6732322822800348 ; i55 = 0.39879144015217155; i56 = 0.7818060791953839; i57 = 0.9797009201839757 i58 = 1.0707852659211583 ; i59 = 0.7342769497222208 ; i60 = 0.2960528761737631; i61 = 0.6419329634235106 i62 = 0.8938441075661789 ; i63 = 1.2082436279312632 ; i64 = 0.8573401904578074; i65 = 0.2438588716389498 i66 = 0.5518544586784879 ; i67 = 0.8186245781022927 ; i68 = 1.1172014750623287; i69 = 1.4864750156733657 i70 = -0.3074083196094299; i71 = 1.5434221599024214 ; i72 = 1.7538814225756034; i73 = -0.07368461229023889 # Notice that i70, i73 are negative. </syntaxhighlight> =====Check all conditions===== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", """+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 = e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69""", """+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 = e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14""", """+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 = e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14""", """+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 = e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14""", """+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 = e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14""" ] </syntaxhighlight> Two Auxiliary Functions: <syntaxhighlight lang=python> def sum_zero (values, tolerance = None) : ''' This function calculates sum of all values in input. If sum is very close to zero, sum is returned as zero. For example: sum_zero((2, -1.99999999999999999999999)) is returned as 0. ''' if tolerance == None : tolerance = Tolerance sump = sumn = 0 for v in values : if v > 0 : sump += v elif v < 0 : sumn -= v max = (sumn, sump)[sump > sumn] sum = sump - sumn if abs(sum) < tolerance: return type(sum)(0) if abs(sum) < tolerance*max : return type(sum)(0) return sum def convert_str_to_tuple (input_str) : ''' input is str like "+ i35 + i40 + i72 - i14 - i69 - i71" or "+ i35 + i40 + i72 == i14 + i69 + i71" This function returns tuple : (i35, i40, i72, - i14, - i69, - i71) ''' str0 = ' '.join(input_str.split()) if '=' in str0 : if '==' in str0 : splitter = '==' else : splitter = '=' str1,str2 = str0.split(splitter) t1 = convert_str_to_tuple (str1) t2 = convert_str_to_tuple (str2) return t1 + tuple(-v for v in t2) str1 = ';+'.join(str0.split('+')) str2 = ';-'.join(str1.split('-')) str3 = ','.join([ v for v in str2.split(';') if v.strip() ]) return eval(str3 + ',') </syntaxhighlight> Check 83 conditions: <syntaxhighlight lang=python> all_conditions = (iconditions + econditions + more_econditions + extra_test_conditions) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] tolerance = 1e-15 count = errors = 0 for condition in all_conditions : t1 = convert_str_to_tuple (condition) sum = sum_zero(t1, tolerance) count += 1 if sum : sx = 'sum, condition' ; print (sx, '=', eval(sx)) errors += 1 print ('Number of tests =', count) print ('Number of errors =', errors) </syntaxhighlight> <syntaxhighlight> sum, condition = (1.2505552149377763e-12, '+ e48 + e16 - e21 - e53') sum, condition = (1.2505552149377763e-12, '+ e49 + e11 - e16 - e54') sum, condition = (7.958078640513122e-13, '+ e52 + e22 - e27 - e57') sum, condition = (1.1368683772161603e-12, '+ e53 + e17 - e22 - e58') sum, condition = (2.2737367544323206e-12, '+ e54 + e12 - e17 - e59') ...... ...... Number of tests = 83 Number of errors = 21 </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ====Calculate R==== <syntaxhighlight> </syntaxhighlight> Produce matrix corrected: <syntaxhighlight lang=python> print ('\nCreating new matrix with corrections:') matrix1 = [ list(v) for v in matrix ] # Deep copy. # e70, i70 are at position 60 in each row of matrix1. # e73, i73 are at position 63 in each row of matrix1. for row in matrix1 : for p in (60,63) : if row[p] : row[p] *= -1 </syntaxhighlight> Circuit parameters: <syntaxhighlight> values_of_circuit_current = ('i40 + i72 + i35', 'i14 + i71 + i69') values_of_circuit_voltage = ( 'e72 + e46 + e47 + e48 + e70 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e73 + e61 + e62 + e63 + e71', 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14', 'e35 + e36 + e37 + e38 + e39 + e65 + e66 + e67 + e68 + e69', 'e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69', 'e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69', 'e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69', 'e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69', 'e35 + e45 + e46 + e47 + e48 + e49 + e11 + e12 + e13 + e14', 'e35 + e36 + e50 + e51 + e52 + e53 + e54 + e12 + e13 + e14', 'e35 + e36 + e37 + e55 + e56 + e57 + e58 + e59 + e13 + e14', 'e35 + e36 + e37 + e38 + e60 + e61 + e62 + e63 + e64 + e14', 'e72 + e31 - e50 + e37 + e38 + e39 + e73 + e61 - e28 + e57' + ( '- e22 + e53 - e16 + e70 + e10 + e11 + e12 + e13 + e14') ) </syntaxhighlight> Calculate 1664 values of R: <syntaxhighlight lang=python> Values_of_R = [] number_of_columns = 65 for I in range(10, 74): new_line = [ 0 ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = (1) i = (I)/10 new_line[-1] = -i matrix1[-1] = new_line values_of_i = solveMbyN (matrix1) values_of_e = [ i*r for i,r in zip ( values_of_i, values_of_R ) ] ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = values_of_e for Istr in values_of_circuit_current : I = eval(Istr) for Estr in values_of_circuit_voltage : E = eval(Estr) ; R = E/I ; Values_of_R += [R] print ('Number of tests =', len(Values_of_R)) tmp = sorted(Values_of_R) minR = tmp[0] ; maxR = tmp[-1] s1 = 'minR' ; print(s1,'=',eval(s1)) s1 = 'maxR' ; print(s1,'=',eval(s1)) sum = sum_zero((minR, -maxR), 1e-14) s1 = 'sum' ; print(s1,'=',eval(s1)) </syntaxhighlight> <syntaxhighlight> Number of tests = 1664 minR = 723.9285759884408 maxR = 723.9285759884439 sum = 0.0 </syntaxhighlight> This code produced 1664 values of R from minR through maxR, both included. With tolerance of <math>1e-14</math> all values of resistance R = 723.92857598844. <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ==Making the Decimal object== The following function verifies that we are working with Decimal objects. <syntaxhighlight lang=python> import sys import decimal getcontext = decimal.getcontext dD = D = Decimal = decimal.Decimal DT = decimal.DecimalTuple dgt = decimal.getcontext() dgt.prec = 50 def print_error (error, x=None, thisName=None) : """ Prints error derived from sys.exc_info(). """ list1 = error.split(',') v1 = ','.join(list1[1:-1]) if thisName : print (thisName) if x != None : print (' Input =', (str(x))[:60]) print (' ', list1[0]) print (' ', v1) print (' ', list1[-1]) def makeDecimal (x, flag = 0) : ''' output = makeDecimal (x [, flag]) x is a single object convertible to Decimal object. returns Decimal object or returns None on error. ''' thisName = 'makeDecimal (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, dD) : return x if isinstance(x, int) : return dD(x) if isinstance(x, float) : return dD(str(x)) if isinstance(x, str) : try : error = '' output = dD(x) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if isinstance(x, (list,tuple,DT)) : try : error = '' v1,v2,v3 = x if isinstance(v2, (tuple,list)) : output = dD(x) else : output = dD( ( v1, list(v2), v3 ) ) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return None return output if flag : print (' Input not recognized.') return None </syntaxhighlight> ===checkComplex(x)=== This function verifies that the object is a valid complex tuple. <syntaxhighlight lang=python> def checkComplex(x, flag = 0) : """ status = checkComplex(x [, flag]) x must be : (v1,v2,'rectangular') or (v1,v2,'polar') v1,v2 must be type Decimal. """ thisName = 'checkComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) try: error = '' v1,v2,str1 = x isinstance (v1,dD) or ({}['v1 must be type Decimal.']) isinstance (v2,dD) or ({}['v2 must be type Decimal.']) (isinstance (str1,str) and str1) or ({}['str1 must be valid string.']) str1 = str1.lower() if str1 == 'rectangular'[:len(str1)] : pass elif str1 == 'polar'[:len(str1)] : pass else : ({}['str1 must be "rectangular" or "polar".']) except : error = str(sys.exc_info()) if error : if flag : print_error (error) return False return True </syntaxhighlight> ===str_to_complex (input_string)=== <syntaxhighlight lang=python> import re # The re pattern for numbers to be found in string of type complex. # In string like ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' # extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'. # eval allows ' 12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J '. # Conversion to Decimal allows: '__987_65432123____4567890_987___'. # eval does not allow ' 0123 ' but allows ' 2e04 '. # Conversion to Decimal allows: ' 0123 ' digits = '[_0123456789]' integer = '[_0123456789]{1,}' # Matches 1 23 345 4567 56789 exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8 Exponent = '({}){{0,1}}'.format(exponent) # exponent 0 or 1 times. float1 = '({})\.{{0,1}}'.format(integer) # Matches 123 2345. float2 = '{}{{0,}}\.{}'.format(digits, integer) # Matches 123.345 .345 float_ = '(({})|({}))'.format(float2,float1) rvalue = '{}({})'.format(float_,Exponent) rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue) ivalue_signed = '{}[Jj]'.format(rvalue_signed) def str_to_complex (input_string) : """ a,b = str_to_complex (input_string) input_string could be ' 1234567890987654323456E-6 + 987654321234567890987e+2J ' input_string could be ' ' ' ( 12345678909876543234561234567890987.654323456E-6 + 987654321234567890987987654321.23764567890987e+2J ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or ' ' ' - ( ( 12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_' 543123456789054_3123456789054312345_67890543654327453E-1_3J ) ) ' ' ' or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)' or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)' or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)', This function retains precision of a,b """ try : status = 0 isinstance(input_string,str) or ({}['input_string not type str.']) str1 = input_string.strip() cx1 = eval(str1) isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.']) except : status = 1 if status : return # This code removes white lines and comments, if any, at end of each line. new_line = ''' '''[-1:] lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ] lines = [ v for line in lines for parts in [ line.split('#') ] for v in [ parts[0]] ] str1 = ''.join(lines) str1 = ''.join(str1.split()) if isinstance (cx1, (int,float)) : resultr = re.search(rvalue_signed, str1) resultr or ({}['rvalue not found.']) dD1 = dD(resultr[0]) v1 = eval(str(dD1)) if v1 == cx1 : return dD1,dD(0) if v1 == -cx1 : return -dD1,dD(0) ({}['dD1 not recognized.']) # cx1 must be complex. It must contain imaginary value. resulti = re.search(ivalue_signed, str1) resulti or ({}['ivalue not found.']) str2j = resulti[0] ; str2 = str2j[:-1] dD2 = dD(str2) str1 = ''.join(str1.split(str2j)) # cx1 may contain real value. resultr = re.search(rvalue_signed, str1) if resultr : dD1 = dD(resultr[0]) else : dD1 = dD(0) cx2 = complex(dD1,dD2) if cx2.real == cx1.real : pass elif cx2.real == -cx1.real : dD1 = dD1.copy_negate() else : ({}['dD1 Not Recognized.']) if cx2.imag == cx1.imag : pass elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate() else : ({}['dD2 Not Recognized.']) cx3 = complex(dD1, dD2) if cx3 == cx1 : return dD1,dD2 ({}['No match for cx3.']) </syntaxhighlight> ===makeComplex(x)=== <syntaxhighlight lang=python> def makeComplex (x, flag = 0) : ''' result = makeComplex (x[, flag]) Input can be tuple with 1,2 or 3 members. If 1 or 2 members, 'rect' is understood. The one member or single object may be int, float, complex, CompleX or string convertible to int, float or complex. x = makeComplex(4) x = makeComplex((4,)) x = makeComplex(('4',0)) x = makeComplex((4,'0', 'rect')) x = makeComplex(4+0j) x = makeComplex('4+0j') x = makeComplex(('4+0j',)) In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' ) output is always (modulus, phase, "polar") or (real_part, imag_part, "rect") modulus, phase, real_part, imag_part are Decimal objects. On error returns None. ''' thisName = 'makeComplex (x) :' if flag : print () print (thisName) print (' Input x =', type(x), (str(x))[:60]) if isinstance(x, CompleX) : # New class CompleX (note the punctuation.) x.check() return (x.r, x.i, 'rect') if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) try : status = 1 a,b = str_to_complex (x) except : status = 0 if status : return a,b,'rect' try : status = 1 result = makeDecimal(x) isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.']) output = result,dD(0),'rect' except : status = 0 if status : return output try : status = 1 v1, = x # Allow for (( 3+4j ),) except : status = 0 if status : return makeComplex (v1) try : status = 1 ; error = '' if len(x) == 2 : v1,v2 = [ makeDecimal (v) for v in x ] str1 = 'rect' elif len(x) == 3 : v1,v2,str1 = x v1,v2 = [ makeDecimal (v) for v in (v1,v2) ] else : ({}['len(x) not in (2,3)']) output = v1,v2,str1 checkComplex(output) or ({}['output not valid complex.']) if str1[0] in 'rR' : output = v1,v2,'rect' else : output = v1,v2,'polar' except : status = 0 ; error = str(sys.exc_info()) if status : return output if flag : print_error (error) return </syntaxhighlight> =Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation. This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation. ==Latera recta et cetera== "Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic. The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines. When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse. From these values everything about the ellipse may be calculated. For example: {{RoundBoxTop|theme=2}} [[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math>''' </br> </br> Axis : (-0.8)x + (-0.6)y + (9.4) = 0</br> Eccentricity = 0.9</br> </br> Directrix 2 : (0.6)x + (-0.8)y + (2) = 0</br> Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0</br> Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0</br> Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0</br> Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0</br> </br> <math>\text{ID2}</math> = (6.32, 7.24)</br> <math>\text{I2}</math> = (7.204210526315789473684, 6.061052631578947368421)</br> F2 = (8, 5)</br> M = (15.16210526315789473684, -4.54947368421052631579)</br> F1 = (22.32421052631578947368, -14.09894736842105263158)</br> <math>\text{I1}</math> = (23.12, -15.16)</br> <math>\text{ID1}</math> = (24.00421052631578947368, -16.33894736842105263158)</br> </br> P = (20.30821052631578947368, -15.61094736842105263158)</br> Q = (10.53708406832736953616, -8.018239580333420216299)</br> R = (5.984, 3.488)</br> S = (10.016, 6.512)</br> T = (19.78712645798841993752, -1.080707788087632415281)</br> U = (24.34021052631578947368, -12.58694736842105263158)</br> </br> Length of major axis: <math>\text{I1I2}</math> = 26.52631578947368421052</br> Length of minor axis: QT = 11.56255298707631300170</br> Length of latus rectum: RS = PU = 5.04 ]] Consider conic section: <math>1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.</math> This curve is ellipse with random orientation. <syntaxhighlight lang=python> # python code ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse result = calculate_abc_epq(ABCDEF) (len(result) == 2) or 1/0 # ellipse or hyperbola (abc1,epq1), (abc2,epq2) = result a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 (e1 == e2) or 2/0 (1 > e1 > 0) or 3/0 print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1) print ('Equation of ellipse in standard form:') print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) ) </syntaxhighlight> <syntaxhighlight> (1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0 Equation of ellipse in standard form: (-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0 </syntaxhighlight> <syntaxhighlight lang=python> # python code def sum_zero(input) : ''' sum = sum_zero(input) If sum is close to 0 and Tolerance permits, sum is returned as 0. For example: if input contains (2, -1.999999999999999999999) this function returns sum of these 2 values as 0. ''' global Tolerance sump = sumn = 0 for v in input : if v > 0 : sump += v elif v < 0 : sumn -= v sum = sump - sumn if abs(sum) < Tolerance : return (type(Tolerance))(0) min, max = sorted((sumn,sump)) if abs(sum) <= Tolerance*min : return (type(Tolerance))(0) return sum </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Major axis=== <syntaxhighlight lang=python> # axis is perpendicular to directrix. ax,bx = b1,-a1 # axis contains foci. ax + by + c = 0 cx = reduce_Decimal_number(-(ax*p1 + bx*q1)) axis = ax,bx,cx print ( ' Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) ) print ( ' Eccentricity = {}'.format(e1) ) print () print ( ' Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) ) print ( ' Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) ) F1 = p1,q1 # Focus 1. print ( ' F1 : ({}, {})'.format(p1,q1) ) F2 = p2,q2 # Focus 2. print ( ' F2 : ({}, {})'.format(p2,q2) ) # Direction cosines along axis from F1 towards F2: dx,dy = a1,b1 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 if dx : distance_F1_F2 = (p2 - p1)/dx else : distance_F1_F2 = (q2 - q1) if distance_F1_F2 < 0 : distance_F1_F2 *= -1 dx *= -1 ; dy *= -1 # Intercept on directrix1 distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1) ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1 print ( ' Intercept ID1 : ({}, {})'.format(xID1,yID1) ) # # distance_F1_F2 # -------------------- = e # length_of_major_axis # length_of_major_axis = distance_F1_F2 / e1 # Intercept1 on curve distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2 xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ] print ( ' Intercept I1 : ({}, {})'.format(xI1,yI1) ) </syntaxhighlight> <syntaxhighlight> Axis : (-0.8)x + (-0.6)y + (9.4) = 0 Eccentricity = 0.9 Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0 Directrix 2 : (0.6)x + (-0.8)y + (2) = 0 F1 : (22.32421052631578947368, -14.09894736842105263158) F2 : (8, 5) Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158) Intercept I1 : (23.12, -15.16) </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>I2, ID2.</math> {{RoundBoxBottom}} ===Latus rectums=== <syntaxhighlight lang=python> # direction cosines along latus rectum. dlx,dly = -dy,dx # # distance from U to F1 half_latus_rectum # ------------------------------ = ----------------------- = e1 # distance from U to directrix 1 distance_from_F1_to_ID1 # half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1) # latus rectum 1 # Focal chord has equation (afc)x + (bfc)y + (cfc) = 0. afc,bfc = a1,b1 cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1)) print ( ' Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) ) P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum print ( ' Point P : ({}, {})'.format(xP,yP) ) U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum print ( ' Point U : ({}, {})'.format(xU,yU) ) distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt()) print (' Length PU =', distance) print (' half_latus_rectum =', half_latus_rectum) </syntaxhighlight> <syntaxhighlight> Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0 Point P : (20.30821052631578947368, -15.61094736842105263158) Point U : (24.34021052631578947368, -12.58694736842105263158) Length PU = 5.04 half_latus_rectum = 2.52 </syntaxhighlight> {{RoundBoxTop|theme=2}} Techniques similar to above can be used to calculate points <math>R, S.</math> {{RoundBoxBottom}} ===Minor axis=== <syntaxhighlight lang=python> print () # Mid point between F1, F2: M = xM,yM = (p1 + p2)/2, (q1 + q2)/2 print ( ' Mid point M : ({}, {})'.format(xM,yM) ) half_major = length_of_major_axis / 2 half_distance = distance_F1_F2 / 2 # half_distance**2 + half_minor**2 = half_major**2 half_minor = ( half_major**2 - half_distance**2 ).sqrt() length_of_minor_axis = half_minor * 2 Q = xQ,yQ = xM + dlx*half_minor, yM + dly*half_minor T = xT,yT = xM - dlx*half_minor, yM - dly*half_minor print ( ' Point Q : ({}, {})'.format(xQ,yQ) ) print ( ' Point T : ({}, {})'.format(xT,yT) ) print (' length_of_major_axis =', length_of_major_axis) print (' length_of_minor_axis =', length_of_minor_axis) # # A basic check. # length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2) # # length_of_minor_axis**2 # ----------------------- = 1-e**2 # length_of_major_axis**2 # # length_of_minor_axis**2 # ----------------------- + (e**2 - 1) = 0 # length_of_major_axis**2 # values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1 sum_zero(values) and 3/0 aM,bM = a1,b1 # Minor axis is parallel to directrix. cM = reduce_Decimal_number(-(aM*xM + bM*yM)) print ( ' Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) ) </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> Mid point M : (15.16210526315789473684, -4.54947368421052631579) Point Q : (10.53708406832736953616, -8.018239580333420216299) Point T : (19.78712645798841993752, -1.080707788087632415281) length_of_major_axis = 26.52631578947368421052 length_of_minor_axis = 11.56255298707631300170 Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0 </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> ===Checking=== {{RoundBoxTop|theme=2}} All interesting points have been calculated without using equations of any of the relevant lines. However, equations of relevant lines are very useful for testing, for example: * Check that points <math>ID2, I2, F2, M, F1, I1, ID1</math> are on axis. * Check that points <math>R, F2, S</math> are on latus rectum through <math>F2.</math> * Check that points <math>Q, M, T</math> are on minor axis through <math>M.</math> * Check that points <math>P, F1, U</math> are on latus rectum through <math>F1.</math> Test below checks that 8 points <math>I1, I2, P, Q, R, S, T, U</math> are on ellipse and satisfy eccentricity <math>e = 0.9.</math> <math></math> <math></math> {{RoundBoxBottom}} <syntaxhighlight lang=python> t1 = ( ('I1'), ('I2'), ('P'), ('Q'), ('R'), ('S'), ('T'), ('U'), ) for name in t1 : value = eval(name) x,y = [ reduce_Decimal_number(v) for v in value ] print ('{} : ({}, {})'.format((name+' ')[:2], x,y)) values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum_zero(values) and 3/0 # Relative to Directrix 1 and Focus 1: distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt() distance_to_directrix1 = a1*x + b1*y + c1 e1 = distance_to_F1 / distance_to_directrix1 print (' e1 =',e1) # Raw value is printed. # Relative to Directrix 2 and Focus 2: distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt() distance_to_directrix2 = a2*x + b2*y + c2 e2 = distance_to_F2 / distance_to_directrix2 e2 = reduce_Decimal_number(e2) print (' e2 =',e2) # Clean value is printed. </syntaxhighlight> {{RoundBoxTop|theme=2}} Note the differences between "raw" values of <math>e_1</math> and "clean" values of <math>e_2.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> I1 : (23.12, -15.16) e1 = -0.9000000000000000000034 e2 = 0.9 I2 : (7.204210526315789473684, 6.061052631578947368421) e1 = -0.9 e2 = 0.9 P : (20.30821052631578947368, -15.61094736842105263158) e1 = -0.9 e2 = 0.9 Q : (10.53708406832736953616, -8.018239580333420216299) e1 = -0.9000000000000000000002 e2 = 0.9 R : (5.984, 3.488) e1 = -0.9000000000000000000003 e2 = 0.9 S : (10.016, 6.512) e1 = -0.9000000000000000000003 e2 = 0.9 T : (19.78712645798841993752, -1.080707788087632415281) e1 = -0.8999999999999999999996 e2 = 0.9 U : (24.34021052631578947368, -12.58694736842105263158) e1 = -0.9 e2 = 0.9 </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> ==Traditional definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. ]] Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant. In the diagram the two fixed points are the foci, Focus 1 or <math>F_1</math> and Focus 2 or <math>F_2.</math> Distance between <math>F_1</math> and <math>F_2</math>, distance <math>F_1F_2</math>, must be non-zero. Point <math>G</math> on perimeter of ellipse moves so that sum of distance <math>F_1G</math> and distance <math>F_2G</math> is constant. Points <math>T_1</math> and <math>T_2</math> are on axis of ellipse and the same rule applies to these points. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> is constant. distance <math>F_1T_1</math> + distance <math>T_1F_2</math> <math>=</math> distance <math>F_1G</math> + distance <math>F_2G</math> <math>=</math> distance <math>F_2T_2</math> + distance <math>T_1F_2</math> <math>= \text{length of major axis.}</math> Therefore the constant is <math>\text{length of major axis}</math> which must be greater than distance <math>F_1F_2.</math> From information given, calculate eccentricity <math>e</math> and equation of one directrix. Choose directrix 1 <math>dx1</math> associated with focus F1. <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> {{RoundBoxBottom}} ==Ellipse at origin== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.''' </br> Sum of distance <math>F_1P</math> and distance <math>F_2P</math> is constant. ]] Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant. By definition distance <math>F_2P</math> + distance <math>F_1P</math> is constant. <math>\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)</math> Expand <math>(1)</math> and result is <math>Ax^2 + By^2 + F = 0\ \dots\ (2)</math> where: <math>A = 4k^2 - 16p^2</math> <math>B = 4k^2</math> <math>F = 4k^2p^2 - k^4</math> When <math>y = 0,</math> point <math>B,\ Ax^2 = -F</math> <math>x^2 = \frac{-F}{A}</math> <math>= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}</math> <math>=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.</math> Therefore: <math>x = \frac{k}{2} = a</math> <math>k = \text{length of major axis.}</math> By definition, distance <math>F_2A</math> + distance <math>F_1A = k.</math> Therefore distance <math>F_1A = a.</math> Intercept form of ellipse at origin: <math>(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2</math> <math>\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1</math> <math>\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1</math> <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> <math></math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Second definition of ellipse== {{RoundBoxTop|theme=2}} [[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where <math>a,b = 20,12</math>.''' </br> At point <math>B,\ \frac{u}{v} = e.</math> </br> At point <math>A,\ \frac{a}{t} = e.</math> ]] Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. Let <math>\frac{p}{a} = e</math> where: * <math>p</math> is non-zero, * <math>a > p,</math> * <math>a = p + u.</math> Therefore, <math>1 > e > 0.</math> Let directrix have equation <math>x = t</math> where <math>\frac{a}{t} = e.</math> At point <math>B:</math> <math>\frac{p}{p+u} = \frac{p+u}{p+u+v} = e</math> <math>(p+u)^2 = p(p+u+v)</math> <math>pp + pu + pu + uu = pp + pu + pv</math> <math>pu + uu = pv</math> <math>u(p + u) = pv</math> <math>\frac{u}{v} = \frac{p}{p+u} = e</math> <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)</math> Statement <math>(3)</math> is true at point <math>A</math> also. Section under "Proof" below proves that statement (3) is true for any point <math>P</math> on ellipse. {{RoundBoxBottom}} ===Proof=== {{RoundBoxTop|theme=2}} [[File:0902ellipse00.png|thumb|400px|'''Proving that <math>\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e</math>.''' </br> Graph is part of curve <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.</math> </br> distance to Directrix1 <math>= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.</math> </br> base = <math>x - p = x - ae</math> </br> <math>\text{(distance to Focus1)}^2 = \text{base}^2 + y^2</math> ]] As expressed above in statement <math>3,</math> second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant. This section proves that this definition is true for any point <math>P</math> on the ellipse. At point <math>P:</math> <math>(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0</math> <math>y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}</math> <math>= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}</math> <math>= e^2x^2 - x^2 + a^2 - a^2e^2</math> base <math>= x-p = x-ae</math> <math>(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2</math> <math>= a^2 - 2aex + e^2x^2</math> <math>= (a-ex)^2</math> <math>\text{distance to Focus1} = \text{distance}\ F_1P = a - ex</math> <math>\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}</math> <math>\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}</math> <math>= (a - ex)\frac{e}{(a-ex)}</math> <math>= e</math> Similar calculations can be used to prove the case for Focus2 <math>(-p, 0)</math> and Directrix2 <math>(x = -t)</math> in which case: <math>\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}</math> <math>= (a + ex)\frac{e}{(a + ex)}</math> <math>= e</math> Therefore: <math>\frac{\text{distance to focus}}{\text{distance to directrix}} = e</math> where <math>1 > e > 0.</math> Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity <math>e.</math> <math></math> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ==Heading== ===Properties of ellipse=== {{RoundBoxTop|theme=2}} [[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.''' </br> </br> Traditional definition of ellipse: </br> <math>\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.</math> </br> </br> Second definition of ellipse: </br> <math>\frac{\text{distance } AF_1} {\text{distance } AG } = \text{eccentricity } e.</math> </br> </br> Triangle <math>A F_1 G</math> is right triangle. </br> <math>e = \cos \angle O F_1 A = \cos \angle F_1 A G</math> ]] Ellipse in diagram has: * Two foci: <math>F_1\ (p,0),\ F_2\ (-p,0).</math> * Length of major axis <math>= \text{distance } I_2 I_1 = 2a</math> * Length of minor axis <math>= \text{distance } A B = 2b</math> * Equation: <math>\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1</math> * Length of latus rectum <math>= \text{distance } P Q</math> * Distance between directrices <math>= \text{distance } D_2 D_1 = 2t</math> Properties of ellipse: * <math>\frac{\text{length of major axis}} {\text{distance between directrices}} = e</math> * <math>\frac{\text{distance between foci}} {\text{length of major axis}} = e</math> * <math>\frac{\text{distance between foci}} {\text{distance between directrices}}= e^2</math> * <math>(\frac{\text{length of minor axis}} {\text{length of major axis}})^2 + e^2 = 1</math> * <math>\frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> * line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ====Major axis==== From traditional definition of ellipse: Distance <math>AF_2\ +</math> distance <math>AF_1</math> = distance <math>I_1F_1\ +</math> distance <math>I_1F_2</math> = distance <math>I_2F_2\ +</math> distance <math>I_2F_1</math> = <math>k.</math> Therefore: Length of major axis = distance <math>I_2I_1 = 2a = k.</math> Distance <math>AF_1 = \frac{k}{2} = a.</math> From second definition of ellipse: <math>\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e</math> <math>= \frac{\text{distance }OI_1}{\text{distance }OD_1}.</math> <math>\frac{\text{length of major axis}}{\text{distance between directrices}} = e.</math> ====Foci==== From second definition of ellipse: <math>\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.</math> <math>a - p = te - ae</math> <math>a - p = a - ae</math> Therefore: <math>p = ae</math> or <math>\frac{p}{a} = e.</math> <math>\frac{\text{distance between foci}}{\text{length of major axis}} = e.</math> <math>\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.</math> ====Minor axis==== Triangle <math>AOF_1</math> is right triangle. <math>\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1</math> <math>= (\frac{b}{a})^2 + (\frac{p}{a})^2 </math> <math>= (\frac{b}{a})^2 + (\frac{ae}{a})^2 </math> <math>= (\frac{b}{a})^2 + e^2 = 1</math> <math>( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1</math> Triangles <math>AOF_1,\ AF_1G</math> are similar. Triangle <math>AF_1G</math> is right triangle. <math>e = \cos \angle OF_1A = \cos \angle F_1AG.</math> ====Latus rectum==== From second definition of ellipse: <math>\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e</math> <math>\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).</math> <math>\frac{\text{distance }PF_1} {a} = 1 - e^2.</math> <math> \frac{\text{length of latus rectum}} {\text{length of major axis}} + e^2 = 1</math> ====Slope of curve==== Curve has equation: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math><math></math> <math>= \frac{-x(1-e^2)}{y}</math><math></math> At point <math>P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}</math> <math>= \frac{ae}{a} = e.</math><math></math> Slope of line <math>PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.</math><math></math><math></math> <math>m_1 = m_2.</math> Therefore line <math>PD_1</math> is tangent to curve at latus rectum, point <math>P.</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> ===Intercept form of equation=== <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1</math> <math></math> <math></math> {{RoundBoxTop|theme=2}} [[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.''' </br> </br> </br> </br> Equation of ellipse has format <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math> where: </br> </br> <math>\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40</math> </br> <math>\text{Length of minor axis} = 2b = \text{distance}\ BA = 24</math> </br> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> </br> <math>\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}</math> </br> </br> <math>e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.</math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> ]] In diagram: Intercept <math>I_1</math> has coordinates <math>(a,0).</math> Intercept <math>I_2</math> has coordinates <math>(-a,0).</math> Intercept <math>A</math> has coordinates <math>(0,b).</math> Intercept <math>B</math> has coordinates <math>(0,-b).</math> Focus <math>F_1</math> has coordinates <math>(f,0)</math> where <math>f = ea.</math> Focus <math>F_2</math> has coordinates <math>(-f,0).</math> Curve has equation <math>\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,</math> called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient. Standard form of this equation is: <math>(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.</math> While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation. Slope of curve: <math>b^2x^2 + a^2y^2 = a^2b^2</math> Derivative of both sides: <math>b^22x + a^22yy' = 0</math> <math>y' = \frac{-xb^2}{ya^2}</math> <math>= \frac{-x(1-e^2)}{y}</math> At point <math>P</math> on latus rectum <math>PQ:</math> <math>m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e</math> Slope of line <math>PD = m_2 = \frac{PF_1}{F_1D} = e</math> <math>m_1 = m_2.</math> Line <math>PD</math> is tangent to curve at latus rectum, point <math>P.</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math>\text{ }</math> <math></math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} ===Example=== {{RoundBoxTop|theme=2}} [[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.''' </br> M is mid-point between F1 and F2. </br> Point R is on minor axis. </br> </br> <math>\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}</math> <math>= e</math> </br> </br> <math>= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}</math> </br> </br> <math>\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}</math> </br> </br> <math>\text{F1:}\ (1, -7)</math> </br> <math>\text{F2:}\ (-1.24, 0.68)</math> </br> length_of_major_axis = 10 </br> <math>\text{M:}\ (-0.12, -3.16)</math> </br> length_of_minor_axis = 6 </br> <math>\text{R:}\ (2.76, -2.32)</math> </br> <math>e = 0.8</math> </br> <math>\text{D1:}\ (1.63, -9.16)</math> </br> <math>\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0</math> </br> <math>\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)</math> </br> <math>\text{epq}\ =\ (0.8,\ 1,\ -7)</math> ]] Given: <syntaxhighlight lang=python> # python code F1 = 1, -7 # Focus 1 F2 = -1.24, 0.68 # Focus 2 length_of_major_axis = 10 </syntaxhighlight> Calculate equation of ellipse. <syntaxhighlight lang=python> F1 = p1,q1 = [ dD(str(v)) for v in F1 ] # Focus 1 F2 = p2,q2 = [ dD(str(v)) for v in F2 ] # Focus 2 length_of_major_axis = dD(length_of_major_axis) half_major_axis = length_of_major_axis / 2 # Direction cosines from F1 to F2 dx = p2-p1 ; dy = q2-q1 divider = (dx**2 + dy**2).sqrt() dx,dy = [ (v/divider) for v in (dx,dy) ] # F2 # p2 = p1 + dx*distance_F1_F2 # q2 = q1 + dy*distance_F1_F2 distance_F1_F2 = (q2-q1)/dy half_distance_F1_F2 = distance_F1_F2 / 2 # The mid-point M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2 # Eccentricity: e = distance_F1_F2 / length_of_major_axis # distance from point R to F1 half_major_axis # ------------------------------------ = e = ----------------------------------------- # distance from point R to Directrix 1 distance from point M to Directrix 1 distance_from_point_M_to_dx1 = half_major_axis / e # Intersection of axis and directrix 1 D1 = xM-dx*distance_from_point_M_to_dx1, yM-dy*distance_from_point_M_to_dx1 D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ] # Equation of Directrix 1 # dx1 = adx1,bdx1,cdx1 adx1,bdx1 = dx, dy # Perpendicular to axis. # adx1*x + bdx1*y + cdx1 = 0 # Directrix 1 contains point D1 cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) ) abc = adx1,bdx1,cdx1 epq = e,p1,q1 ABCDEF = ABCDEF_from_abc_epq (abc,epq, 1) </syntaxhighlight> Equation of ellipse in standard form: <math>(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0</math> For more insight into method of calculation and proof: <syntaxhighlight lang=python> if 1 : print ('F1: ({}, {})'.format(p1,q1)) print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1)) print ('F2: ({}, {})'.format(p2,q2)) print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2)) print ('length_of_major_axis =', length_of_major_axis) print ('M: ({}, {})'.format(xM,yM)) print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM)) # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2 half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt() length_of_minor_axis = half_minor_axis * 2 s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1)) # Direction cosines on major axis: print ('dx,dy =', dx,dy) # Direction cosines on minor axis: dnx,dny = dy,-dx print ('dnx,dny =', dnx,dny) # One point on minor axis: R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis print ('R: ({}, {})'.format(xR,yR)) print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR)) # Verify that point R is on ellipse: sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0 s1 = 'e' ; print (s1, '=', eval(s1)) print ('D1: ({}, {})'.format(xD1,yD1)) print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1)) print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1)) print() # For proof, reverse the process: (abc1,epq1), (abc2,epq2) = calculate_abc_epq (ABCDEF) a1,b1,c1 = abc1 ; e1,p1,q1 = epq1 print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1)) print ('Eccentricity e1: {}'.format(e1)) print ('F1: ({}, {})'.format(p1,q1)) print() a2,b2,c2 = abc2 ; e2,p2,q2 = epq2 print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2)) print ('Eccentricity e2: {}'.format(e2)) print ('F2: ({}, {})'.format(p2,q2)) print ('\nEquation of ellipse with integer coefficients:') A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ] str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0' print (str1.format(A,B,C,D,E,F)) </syntaxhighlight> <syntaxhighlight> F1: (1, -7) F1: (x - (1))^2 + (y - (-7))^2 = 1 F2: (-1.24, 0.68) F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1 length_of_major_axis = 10 M: (-0.12, -3.16) M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1 length_of_minor_axis = 6 dx,dy = -0.28 0.96 dnx,dny = 0.96 0.28 R: (2.76, -2.32) R: (x - (2.76))^2 + (y - (-2.32))^2 = 1 e = 0.8 D1: (1.63, -9.16) D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1 Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0 Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0 Eccentricity e1: 0.8 F1: (1, -7) Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 Eccentricity e2: 0.8 F2: (-1.24, 0.68) Equation of ellipse with integer coefficients: </syntaxhighlight> <math>(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0</math> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <math></math> <math></math> <math></math> <math></math> <math></math> {{RoundBoxBottom}} =allEqual= {{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}} <div style="padding-top:0.25em; 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We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> =====Welcomen===== {{Robelbox|title=|theme={{{theme|9}}}}} <div style="padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em; background-color: #FFFFFF; "> [[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]]. </div> <syntaxhighlight lang=python> # python code. if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 : pass </syntaxhighlight> {{Robelbox/close}} {{Robelbox/close}} {{Robelbox/close}} <noinclude> [[Category: main page templates]] </noinclude> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>26</code> || <code></code><code>419</code> |} =Testing= ======table1====== {|style="border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;" align="center" | Hello As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 </syntaxhighlight> |} {{RoundBoxTop|theme=2}} [[File:0410cubic01.png|thumb|400px|''' Graph of cubic function with coefficient a negative.''' </br> There is no absolute maximum or absolute minimum. ]] Coefficient <math>a</math> may be negative as shown in diagram. As <math>abs(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^3.</math> When <math>x</math> has a very large negative value, <math>f(x)</math> is always positive. When <math>x</math> has a very large positive value, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} <math>x_{poi} = -1</math> <math></math> <math></math> <math></math> <math></math> =====Various planes in 3 dimensions===== {{RoundBoxTop|theme=2}} <gallery> File:0713x=4.png|<small>plane x=4.</small> File:0713y=3.png|<small>plane y=3.</small> File:0713z=-2.png|<small>plane z=-2.</small> </gallery> {{RoundBoxBottom}} <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight lang=python> </syntaxhighlight> <syntaxhighlight> </syntaxhighlight> <syntaxhighlight> 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727 3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147 0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989 6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471 6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723 5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162 0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342 1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698 6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112 0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472 </syntaxhighlight> <math>\theta_1</math> {{RoundBoxTop|theme=2}} [[File:0422xx_x_2.png|thumb|400px|''' Figure 1: Diagram illustrating relationship between <math>f(x) = x^2 - x - 2</math> and <math>f'(x) = 2x - 1.</math>''' </br> ]] {{RoundBoxBottom}} <math>O\ (0,0,0)</math> <math>M\ (A_1,B_1,C_1)</math> <math>N\ (A_2,B_2,C_2)</math> <math>\theta</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} (6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\ p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math>\begin{align} 1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}</math> <math>\ \ \ \ \ \ \ \ </math> :<math> 4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 </math> ozk3jgy14fqy5jngbubgsosnnhtzjx4 0 235272 2834891 2834753 2026-09-28T19:59:17Z The Citer 3110681 /* Polypodiophyta (Monilophyta) */ 2834891 wikitext text/x-wiki [[Image:Diversity of plants (Streptophyta) version 2.png|thumb|300px|right|A sample of plant diversity.]] In botany, the equivalent of a Phylum is called a division. The Kingdom Plantae is divided into 13 Divisions. A Division (pl. Phyla) is the largest formal major grouping within plant taxonomy below kingdoms. This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement. Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Plant Divisions. Other sources may combine or split these listed Divisions. However, at this time, the list presented here should stand in good stead for an introduction to the topic of plant diversity. There are approximately 380,000 plant species that have been described by science. This list tries to give the following information on each Division: *Division Name *A link to a subpage discussing that Phylum in more detail (if it yet exists) *Name Meaning (in English) *An English Common Name, where one is in regular use *Distinguishing characteristics of plants within the Division *An approximate number of species described within that Division. Since botany does not stand still, this number can change. You can also see [[Introduction to Taxonomy]] for more on that topic. ==Anthocerotophyta== [[Image:Hornwort (3144429129).jpg|thumb|100px|right|Hornworts.]] [[/Anthocerotophyta/]] Name Meaning: Anthoceros-like plant English Common Name: Hornworts Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system Approximate number of species described: 100-300 or more ==Bryophyta== [[Image:Mose09.jpg|thumb|100px|right|Moss, a bryophyte.]] [[/Bryophyta/]] Name Meaning: Bryum-like plant, moss plant English Common Name: Moss Major distinguishing characteristics: Persistent branched sporophytes, no vascular system Approximate number of species described: 12,000 ==Charophyta== [[Image:Chara overview.jpg|thumb|100px|right|''Chara'', a Charophyte.]] [[/Charophyta/]] Name Meaning: Chara-like plant English Common Name: Charophytes Major distinguishing characteristics: Approximate number of species described: 1,000 ==Chlorophyta== [[image:Bulletin de l'Acadmie impriale des sciences de St.-Ptersbourg (20431048865).jpg|thumb|100px|right|A Chlorophyte.]] [[/Chlorophyta/]] Name Meaning: Yellow-green plant English Common Name: Chlorophytes Major distinguishing characteristics: mainly autotrophs with exceptions and have the same chlorophyll a and b pigments as "higher" plant divisions Approximate number of species described: 8,000 ==Cycadophyta== [[Image:Unidentified cycad in greenhouse.jpg|thumb|100px|right|Unidentified cycad in greenhouse.]] [[/Cycadophyta/]] Name Meaning: Cycas-like plant, palm-like plant English Common Name: Cycads Major distinguishing characteristics: Seeds, crown of compound leaves Approximate number of species described: 100 - 200 ==Ginkgophyta== [[Image:Gingko biloba2.jpg|thumb|100px|right|''Gingko biloba''.]] [[/Ginkgophyta/]] Name Meaning: Ginkgo-like plant English Common Name: Ginkgo, maidenhair tree Major distinguishing characteristics: Seeds not protected by fruit Approximate number of species described: 1 living, about 50 extinct ==Glaucophyta== [[Image:Glaucophyte.jpg|thumb|100px|right|A glaucophyte.]] [[/Glaucophyta/]] Name Meaning: Blue-green plant English Common Name: Glaucophytes Major distinguishing characteristics: Approximate number of species described: 14 - 26<ref>[http://jcs.biologists.org/content/131/2/jcs203414 The monoplastidic bottleneck in algae and plant evolution | Journal of Cell Science]</ref> ==Gnetophyta== [[Image:Gnetum scandens (6780786863).jpg|thumb|100px|right|''Gnetum scandens''.]] [[/Gnetophyta/]] Name Meaning: Gnetum-like plant English Common Name: Gnetophytes Major distinguishing characteristics: Seeds and woody vascular system with vessels. Approximate number of species described: 70 ==Lycopodiophyta (Lycophyta)== [[Image:Clubmoss - Flickr - pellaea (1).jpg|thumb|100px|right|Clubmoss.]] [[/Lycopodiophyta (Lycophyta)/]] Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living ==Magnoliophyta (Anthophyta)== [[Image:Sweetbay Magnolia Magnolia virginiana Flower Closeup 2242px.jpg|thumb|100px|right|''Magnolia virginiana''.]] [[Plant Divisions (Phyla)/Magnoliophyta|Magnoliophyta]] Name Meaning: Magnolia-like plant English Common Name: Flowering plants, angiosperms Major distinguishing characteristics: Flowers and fruit, vascular system with vessels Approximate number of species described: 300,000 ==Marchantiophyta (Hepatophyta)== [[Image:Liverwort Ferndale Park.jpg|thumb|100px|right|Liverwort.]] [[Plant Divisions (Phyla)/Marchantiophyta|Marchantiophyta (Hepatophyta)]] Name Meaning: Marchantia-like plant, liver plant English Common Name: Liverworts Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system Approximate number of species described: 9,000 ==Pinophyta (Coniferophyta)== [[Image:Taxus wallichiana kz1.jpg|thumb|100px|right|''Taxus wallichiana'', the Himalayan Yew, a conifer.]] [[/Pinophyta (Coniferophyta)/]] Name Meaning: Pinus-like plant, cone-bearing plant English Common Name: Conifers Major distinguishing characteristics: Cones containing seeds and wood composed of tracheids Approximate number of species described: 629 extant ==Polypodiophyta (Monilophyta)== [[Image:Tree Fern.jpg|thumb|100px|right|Tree fern fronds and fiddleneck (growing young frond).]] [[/Polypodiophyta/]] (Monophyte) Once called [[wikipedia:Fern|Pteridophyta]] (outdated! The sub-divisions Lycopodiophyte and Euphyllophyte have been differentiated) Name Meaning: Many foot plant, Polypodium-like plant English Common Name: ferns, horsetails Major distinguishing characteristics: Prothallus gametophytes and vascular system Approximate number of species described: 9000 ==Other Resources== *[http://tolweb.org/Green_plants/2382 Tree of Life, Green Plants] *[http://eol.org/pages/281/overview Encyclopedia of life, Plantae] *[[Animal Phyla]], a companion piece to this one *[[Streptophytes]], a "slice" of this page.* ==Quizzes== [[Plant Divisions (Phyla)/Magnoliophyta/Quiz]] ==References== * [[Wikipedia:Phylum]] {{reflist}} [[Category:Plants]] [[Category:Taxonomy]] [[Category:Botany]] [[Category:Biology]] p5qxyq2ef33n8nsy3mt3zvtl101nk4z 0 264274 2834868 2721273 2026-09-28T17:31:18Z ~2026-52184-98 3111768 /* 31 December 1889, Tuesday */ 2834868 wikitext text/x-wiki [[Social Victorians/Timeline/1840s|1840s]] [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] ||[[Social Victorians/Timeline/1880s | 1880s Headlines]][[Social Victorians/Timeline/1880 | 1880]][[Social Victorians/Timeline/1881 | 1881]][[Social Victorians/Timeline/1882 | 1882]][[Social Victorians/Timeline/1883 | 1883]][[Social Victorians/Timeline/1884 | 1884]][[Social Victorians/Timeline/1885 | 1885]][[Social Victorians/Timeline/1886 | 1886]][[Social Victorians/Timeline/1887 | 1887]][[Social Victorians/Timeline/1888 | 1888]] 1889 || [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]] ==Sometime in 1889== A. H. Macmurdo purchased 20 Fitzroy, London, and redecorated it memorably. From the 1 January 1890 ''Morning Post'': <blockquote>Equally the attempt of Mr. Gladstone to invent a brand new system of "electoral statistics," directed to the conclusion that he had in the minority the leadership of the only true majority, with the implication that the Unionists, if honest men, would immediately resign their trust, has been received with good-humoured incredulity.<ref>''Morning Post'' 1 January 1890, Wednesday (No. 36,675): p. 4, Col. 7. '''British Newspaper Archive'''.</ref></blockquote> ==January 1889== ===1 January 1889, Tuesday, New Year's Day=== === 16 January 1889 === A Jack the Ripper letter "refers to 'my trip to Bradford'" (Cornwell 296). === 22 January 1889, Tuesday === ==== Wedding of Lady Eleanor Lambton and Lord Robert Cecil ==== The [[Social Victorians/Cecil Lambton Wedding 1889 January 22|wedding of Lady Eleanor Lambton and Lord Robert Cecil]] took place at 2:30, St. George's Church, Hanover Square, London. ==February 1889== ==March 1889== ===6 March 1889, Wednesday=== Ash Wednesday. ==April 1889== ===10 April 1889, Wednesday=== According to the summary of the weather in the 1 January 1890 Morning Post, there was "intense darkness from 10 a.m. to 2 p.m." "The Weather of 1889."<ref>''The Morning Post'', 1 January 1890 (No. 36,675): p. 6, Col. 3.</ref> ===11 April 1889, Thursday=== According to the summary of the weather in the 1 January 1890 Morning Post, the "intense darkness" of the day before returned "for a short time next day" <cite>"The Weather of 1889." The Morning Post, 1 January 1890 (No. 36,675): page 6 Col. C.</cite> ===12 April 1889, Friday=== Amy Levy and Elizabeth Pennell both attended what Pennell called a "converzazione" at the Fabian Society in the Bloomsbury Town Hall. Pennell says, "We went to the Converzazione of the Fabian Society in Bloomsbury Town Hall. A collection of cranks, native and foreign: young women in extraordinary costumes, one a perfect Burne-Jones, played the violin; young men with long hair and velvet coats. Most people were in evening dress so that a conspicuous figure was [[Social Victorians/People/George Bernard Shaw|George Bernard Shaw]] in grey Jaeger get-up, flirting outrageously with all the girls in the room" (Linda Hunt Beckman, Amy Levy: Life and Letters, p. 179; posting on victoria@listserv.indiana.edu Linda Hunt Beckman <beckman@temple.edu>, 20 October 2003, "Re: conversaziones at Berkeley Galleries; mourning customs"). ===19 April 1889, Friday=== Good Friday. ===21 April 1889, Sunday=== Easter Sunday. ===28 April 1889, Sunday=== Jack Yeats <quote>attended the funeral of the Duchess of Cambridge, whose husband was Queen Victoria's uncle, and commander-in-chief of the British Army. She was a very popular woman; Jack described the event as 'more like a race meeting than a funeral'. Street touts sold 'memorium cards of the dear old Dutchie', and charged fourpence for standing room at the curbside. Griffin was with Jack</quote> (Bruce Arnold. Jack Yeats. p. 46) ==May 1889== Sometime in May 1889 the The Inner Order of the Golden Dawn met at the offices of The Sanitary Wood Wool Co. (Howe 85 31). Dinner planned in honor of James Whistler, with a long list of celebrities invited, including [[Social Victorians/People/Arthur Collins|Arthur Collins]], Equerry to the Queen, William Christian Symons, the recipient of a letter from Whistler's son,<blockquote>Sir Coutts Lindsay (1824-1913), Bart., co-founder of the Grosvenor Gallery ..., and Edmond, Prince de Polignac (1834-1901), composer ...; diplomats, such as James Rennell Rodd (1858-1941), 1st Baron Rennell, poet and diplomat ...; lawyers, including Sir John Charles Sigismund Day (1826-1908), QC, collector ..., and Sir George Henry Lewis (1833-1911), society lawyer ...; and scientists, including Edwin Ray Lankester (1847-1929), Professor of Zoology and Comparative Anatomy .... There are several men involved with drama and music, such as John Hollingshead (1827-1904), journalist and manager of the Gaiety Theatre ..., Richard D'Oyly Carte (1844-1901), impresario and property developer ..., and Pablo de Sarasate y Navascues (1844-1908), violinist ...; writers, such as Sheridan Ford (1860-1922), poet, critic, politician and writer on art ..., Stéphane Mallarmé (1842-1898), Symbolist writer and poet ..., and Comte Robert de Montesquiou-Fezensac (1855-1921), Symbolist writer and poet, and collector [more]. Art critics and editors include Theodore Child (1846-1892), journalist and art critic ..., Théodore Duret (1838-1927), art critic and collector ..., Henry Du Pré Labouchère (1831-1912), journalist and Liberal MP ...,William Michael Rossetti (1829-1919), civil servant and critic ..., and George Augustus Sala (1828-1895), artist, journalist and critic .... Nor surprisingly, there are many patrons and collectors, such as Alexander ('Aleco') Ionides (1840-1898), businessman ..., Alfred Chapman (1839-1917), engineer and collector ..., and Wickham Flower (b. ca 1836), solicitor and collector ...; some art dealers, such as Charles William Deschamps (1848-1908), art dealer ...; and, of course, many artists, including Joseph Edgar Boehm (1834-1890), sculptor ..., Hilaire-Germain-Edgar Degas (1834-1917), artist ..., John Lavery (1856-1941), painter ..., William Quiller Orchardson (1832-1910), genre and portrait painter ..., Théodore Roussel (1847-1926), painter and print-maker ..., Walter Richard Sickert (1860-1942), artist and writer on art ..., Francis ('Frank') Job Short (1857-1945), printer and print-maker ..., and John Singer Sargent (1856-1925), artist .... Finally, there are a few family members, such as Hon Charles Ernest Thynne (1849-1906), solicitor, husband of JW's niece Annie Haden ..., and William McNeill Whistler (1836-1900), physician, JW's brother .... (http://www.whistler.arts.gla.ac.uk/correspondence/people/display/?cid=5635&nameid=Collins_A&sr=0&surname=Collins&firstname=Arthur&rs=1#ms05635)</blockquote> === 18 May 1889, Saturday === Augustus Harris's Italian Opera season opened at Covent Garden with Bizet's ''I Pescatori di Perle'' (which isn't Italian). The reporting is not perfectly clear, but the Princess of Wales, her daughters and the Duchess of Edinburgh do not seem to have been present in 1889, but they were in 1887 for the Italian Opera's first season at Covent Garden.<blockquote>Covent Garden Theatre, on Saturday, when the Italian Opera season was inaugurated by a performance of Bizet's early opera "I Pescatori de Perle," presented the most brilliant appearance imaginable, and being even more noteworthy that the first season's assemblage which Mr. Harris brought together two years ago, The Princess of Wales and the three young Princesses, together with the Duchess of Edinburgh, occupied the Royal box. Among the host of fashionabie people present were Lord and Lady Charles Beresford, the Duke of Portland, Lady Randolph Churchill, Lady Rosebery, the Duchess of Marlborough, Sir Julian Goldsmid, Mr. Harry H. Marks, Lord Wolverton, Mrs. Edward Sassoon, Mrs. Vanderbilt, the Hon. F. E Allsopp, Lord Revelstoke, Mr. H. L W. Lawson, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], Lady Windsor, Mrs. Lawrence, Lady Dudley, the Duchess of Montrose, Mr. Alfred de Rothschild, Baron Ferdinand de Rothschild, Sir Charles Tennant, Sir Phillip Currie, Mr. Arthur Cohen, Mr. Critchett. Mr. Poland, the Hon. G. Curzon, Mr. Seager Hunt, M.P., the Duchess of Leinster, Lord Latham, Mr. and Mrs. Arthur Raphael, and Mr. Corney Grain.<ref>"Royal Italian Opera. A Brilliant Assemblage at the First Night at Covent Garden." ''Evening News'' (London) 20 May 1889, Monday: 2 [of 4], Col. 7b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003283/18890520/026/0002. Same title and p.</ref></blockquote>Paul Rodmell says it was 19 May 1889, but that was a Sunday, so is that an error?<ref>Rodmell, Paul. ''Opera in the British Isles, 1875–1918''. Music in 19th-century Britain. Ashgate, 2013. Appendix 2.1.</ref> === 27 May 1889, Monday, 11 p.m. === ==== The Queen's State Ball at Buckingham Palace ==== It seems to have been hosted by the Prince and Princess of Wales and a number of their children. Queen Victoria was not present. The dancing began at 11 p.m., after the Prince and Princess of Wales and their family and attendants arrived.<ref>"The Queen's State Ball." ''Morning Post'' 28 May 1889, Tuesday: 5 [of 8], Col. 5a–7b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18890528/048/0005. Same print title and p.</ref> ==June 1889== The last meeting of the Men and Women's Club was held in June 1889 (Bland 41). The "initial London performance of 'A Doll's House' was in June 1889 and was attended by Eleanor Marx, Clementina Black, and Amy Levy (Bernstein paragraph 12). Then Walter Besant wrote "The Doll's House -- And After," attacking Nora for the destruction of her family twenty years later. G. B. Shaw wrote another version in 1890, a sequel to the original as well, "Still After the Doll's House," attacking Besant's vision (Time [February 1890]: 197–208). Eleanor Marx and Israel Zangwill then wrote "A Doll's House Repaired," in which the door slamming is the one that locks Nora in (Time [March 1891]: 239–253). ===5 June 1889, Wednesday=== Derby Day at Epsom Downs. According to the ''Morning Post'', <quote>The Derby Day. / Lady George Hamilton's second evening party, at the Admiralty. / Lady Trevelyan's dance, instead of the 27th of May. / Lady Jane Lindsay's first dance. / The Hon. Mrs. Greville Vernon's dance.</quote> ("Arrangements for This Day." The Morning Post Wednesday, 5 June 1889: p. 7 [of 12], Col. 6B). ===6 June 1889, Thursday=== According to the year-end summary of the weather in the 1 Janaury 1890 ''Morning Post'', there was a "grand display of lightning during the violent thunderstorm of June 6" <cite>"The Weather of 1889." The Morning Post, 1 January 1890 (No. 36,675): page 6 Col. C.</cite> === 8 June 1889, Saturday === The Royal Institute of Painters in Water-Colours opened an exhibition of "the works of the 'English Humourists in Art.'"<blockquote>The members of the Royal Institute of Painters in Water-Colours, Piccadilly, W., through their president, Sir James D. Linton, and Mr. James Orrock, R.I., have made arrangements for a representative exhibition of the works of the "English Humourists in Art." The scope will be a wide one, from Hogarth, Rowlandson, etc., through the successive stages of humorous graphic art to its present development, as exemplified in the members of the artistic staff of ''Punch'' and contemporary pictorial humourists. The exhibition opens on the 8th of June, and is under distinguished patronage. The Queen will exhibit original drawings by Rowlandson, etc., from the royal collection; the Duchess of St. Albans, the [[Social Victorians/People/Westminster|Duke of Westminster]], the [[Social Victorians/People/Rosebery|Earl of Rosebery]], the [[Social Victorians/People/Mayo|Earl of Mayo]], the Hon. [[Social Victorians/People/Bourke|Algernon Bourke]], the Hon. Michael Sandys, Mr. [[Social Victorians/People/Henry Irving|Henry Irving]], and other fortunate possessors of original treasures of the requisite kind will contribute to the collection. The leading features of the Exhibition will be the gathering of original works by Rowlandson, an assemblage of the original illustrations designed from the works of Charles Dickens, including the series by George Cruikshank, H. K. Browne (“ Phiz ”), John Leech, Fred Barnard, Charles Green, R.I., etc. A most important series of water-colour drawings by the last-named artist, illustrating episodes from Dickens on a more ambitious scale than has been previously attempted, will be contributed by Mr. William Lockwood, who commissioned Mr. Charles Green to undertake this ''tour de force''. Nearly all the original works in oils by Mr. Fred Barnard will be exhibited. Randolph Caldecott will also be well represented. There will likewise be numerous specimens of Messrs. John Tenniel, R.I., Charles Keene, George Du Maurier, R.W.S., Harry Furniss, Linley Sambourne, Gordon Thompson, Alfred Bryan, J. F. Sullivan; and others. Mr. Gilbert Dalziel is sending a selection from the best examples of the late W. G. Baxter.<ref>"An Exhibition of Humourists." ''St James's Gazette'' 22 May 1889, Wednesday: 8 [of 16], Col. 2a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18890522/041/0008. Print p. 8.</ref></blockquote> ===26 June 1889, Wednesday=== There was apparently a regular celebration of [[Social Victorians/People/Arthur Collins|Arthur Collins]]' birthday, 26 June, by Bret Harte, George Du Maurier, [[Social Victorians/People/Arthur Sullivan|Arthur Sullivan]], Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902. ==July 1889== === 2 July 1889, Tuesday === ==== The Shah at a Covent Garden Opera Performance ==== These two articles are reportage, written in a personal, 1st-person style.<blockquote>The "Arabian Nights" at Covent Garden. I. — By Our Operatic Special. Truly a great function! Worthy of the occasion, and worthy of everybody concerned! Such was the verdict passed upon last night's show at Covent Carden by those who were lucky enough to be present at what was practically a unique performance. State visits to the opera are few and far between, and we must carry our memories back some sixteen years in order to recall a similar combination of royalty, aristocracy, millinery, and jewellery. No one has a better recollection of the operatic past than Mr. Edward Hall, the ''doyen'' of the Covent Garden box-office, who has been connected with the establishmient for over forty years. This veteran told me last night, when all was over, that the brilliancy of the occasion fully equalled, if not surpassed, both the Shah's visit to Bow-street (the theatre, not the police- court), in 1873, and also the Sultan of Turkey's entertainment at the same house in 1867. At any rate the vast majority of us had never seen anything quite like it before, and consequently we were duly astonished and delighted. Arriving at the opera-house shortly before eight o'clock, I devoted myself immediately to making a short tour of inspection of all the arrangements. The entrance-hall was judiciously ornamented with strange and prickly shrubs, and had quite the appearance of a tropical jungle tempered with an allowance of red cloth. Turning to the grand staircase, which was covered with a carpet of peacock-blue, I made my way through leafy masses and nodding blossoms to the crush-room. Here all was cool and inviting. Huge blocks of ice, rendered translucent by artfully concealed coloured lights, filled every corner, mirrors reflected palms of all sorts and sizes, and comfortable lounges were scattered about for the benefit of the Royalties. This "elegant apartment" adjoined the Royal box, and was of course inaccessible to outsiders. The smoking balcony over the portico was also reserved for the various "highnesses," and a long table covered with light refreshments looked quite tempting with its glittering array of glass and plate. However, it was now time to seek the auditorium, and accordingly I hurried downstairs again and entered the stalls. As I gazed round the house a perfect "Arabian Nights" vision burst upon me. Imagine Covent Garden Theatre with every circle lined with bouquets, and every chair decked with a satin programme pinned to it, antirnaccassar fashion! The effect was one to be seen in order to be fully realized. Goodness only knows where the florists found material for all these bunches of roses and carnations, and for the substantial button-holes which were placed at the disposal of the gentlemen; one would almost expect a flower-ramine [?] in the adjacent market as the natural result. The audience was coming in slowly and sedately, and I had ample time and opportunity to look round me and take note of a few well-known faces. Lord and Lady Rothschild were among the earliest arrivals — he in a scarlet coat, and she in white and diamonds; Lady Mandeville took up her position early in Mrs. William Carrington's box. Lord and Lady Charles Beresford and Lord and Lady de Grey had deserted their usual quarters and occupied the Prince of Wales's box, which of course was not required by Royalty. Lady de Grey had on her lovely diamond tiara. Indeed, all over the theatre it positively rained precious stones, and there was hardly a lady on the grand or pit tiers who did not "sport" something magnificent in the way of a "head-piece." Mr. Edward Lawson, and Mr. Augustus Spalding each wandered about the stalls in the red coat of a deputy lieutenant; Mr. Reuben Sassoon wore black court dress, and so did Dr. Pavey, of "Guy's." Mr. Higgins hovered between the stalls and one of the omnibus boxes; Lady Dudley divided herself between her own domain, where she was accompanied by Miss Mordaunt, and the Beresford box; and Mr. Potter, Q.C., wore "common or garden" evening clothes. Others who came in good time were the Danish Minister (in gorgeous array) and Madame De Falbe, wearing her wonderful emeralds; Mr. De Murietta, Lord and Lady Will[i]am Nevill, the latter wearing diamonds, pearls, and sapphires; Miss Murietta, in white, and Mr. Van Raalte, who brought his wife in a charming light blue gown. But stay! Here is Signor Mancinelli in his place, just ready to begin. The Royal party is not expected until nine o'clock, and so the first few items will be got through in their absence. The orchestra plays the overture to "William Tell" splendidly, and then Mdme. Melba comes on to sing the mad scene from "Lucia," under the baton of Signor Randegger. Next Signor Arditi takes his turn with the "stick," and Beethoven's immortal "Leonora No. 3' is given. After this there is a pause; Signor Mancinelli resumes his comnand, and stands, waiting for his signal. It is past nine o'clock, and the Royalties are already overdue. A few minutes of anxious expectation, and then distant cheers are heard by those in the corridors. The audience rise as one being, the conductor's stick makes a significant movement, there is a mighty roll of drums leading into the National Anthem, and then his Majesty the Shah enters the Royal box with the Princess of Wales on his arm. The Prince of Wales follows with Princess Louise, and then come Prince Albert Victor and Princess Victoria of Wales; Prince George and Princess Beatrice; the Earl of Fife, escorting his fiancée, Princess Louise of Wales; and Prince Henry of Battenberg with Princess Maud. The Princess of Wales wears a magnificent flame-coloured brocade, and her famous diamond tiara; Princess Beatrice is in black, and Princess Louise and the three young Princesses are all in white. In attendance on the Royal party are the Persian Minister and several members of the Shah's suite, the Duke of Portland, Lord Radnor, Lord Lathom, Sir Henry Drummond Wolff, Sir Henry Rawlinson, Sir Spencer Ponsonby, Lady Sophia Macnamara, Lady Emily Kingscote, and Miss Cochrane. Every one has a good stare at the Shah while "God save the Queen" and the Persian National Anthem are being played, and then the Royalties settle down in their charming white and yellow ''loge'', and begin to turn over the books of the words, which have been elaborately bound in crimson watered silk. The curtain rises almost immediately on the fourth act of "Faust," which is sung by Mdrne. Albani, Mdme. Scalchi, the Brothers de Reszke, and M. Lassalle. His Oriental Majesty appears to enjoy the music, and waves his right band vigorously, while the Soldiers' Chorus is being sung. Then there is a brief interval, during which I perambulate the house and make up the following "little iist" of notabilities. Lord Limerick; Mrs. Edward Lawson, in white; Mr. and Mrs. Harry Lawson; Mr. Maximilian de Bathe; Mr. and Mrs. Hulse; Mr. George Curzon, M.P., who is in Lord Windsor's box; Lady Colin Campbell, with a lovely feather fan; Mrs. Labouchere, in pink; the Duchess of Newcastle, with her daughter, the Duchesse d'Avigliano; [[Social Victorians/People/Bourke|Mr. and Mrs. Algernon Bourke]], in the Murietta's [sic] box; Mrs. Arthur Wilson, in black, who manages to drop her banquet onto someone's head; Lady Zetland, in white; the Duchess of Portland, also in white; the Duchess of Marlborough; Lady Lucas; Lord and Lady Windsor; Mrs. Parkinson Sharpe and her daughters; Mrs. Lawrence, in black and white; Mr. Charles Lucas; Mrs. Edward Sassoon, in white; Mr. "Arthur Roberts" Nugent, in a bear-skin; Sir Morell Mackenzie, with his German star; Mr. Charles Hall, Q.C.; the Duchess of Manchester, Mr. Arthur Chappell, Mrs. Arthur Paget, and Lady Mary Mills, who looks lovely in grey velvet and a diamond tiara; Mrs. Henry Oppenheim, in red; Lady Rosebery, with Baron Ferdinand de Rothschild; Lord Abington; Mr. and Mrs. Hanbury; Lady Broome, in terra cotta; and Colonel and Mrs. William Carington. In the "omnibus" boxes are Lord Calthorpe, Lord Chesterfield, Mr. Chaplin, Colonel Oliver Montague, Lord Henry Paulet, Mr. Baring, Mr. Williams, and Mr. C. de Murietta. Mrs. Arthur Sassoon is with Lady Forbes; Mrs. McEwan has brought Lady Jane Taylour, and Lady Howe, in white, has come with Lady Hindlip. Amongst others present I see Lady Randolph Churchill; Lady de Trafford, in brown satin, with pale pink, roses; the Spanish Ambassador; Mrs. Albert Brassey; Mrs. Adrian Hope, in white satin; Lady Curzon; Lady Claude Hamilton, in salmon pink; Maria, Marchioness of Ailesbury, in black and gold; Mrs. Tatton Egerton, Mr. Alfred de Rothschild; and a heap of others. The Guards about the theatre make a particularly brilliant show, and the twenty-four Beefeaters, who have been sent over from St. James's Palace, remind one of the "Yeomen of the Guard." The second Part of the programme goes quite as well as the first, and Mdme. Melba makes an especial hit in the valse from "Romeo et Juliette." When all is over the Royal party passes down the staircase and gradually disperses.Then the "Wales" family departs for Marlborough House, and Lord Fife is left to take care of Princess Beatrice instead of his ''fiancée''. When all the Royalties have finally gone every one, [sic] else slowly leaves the big opera house, and I myself, turning my steps southward, run against a little fair man, and have the pleasure of congratulating Mr. Clarkson on having "wigged" three operas in one evening. So ends a memorable function. A Display of Unparalleled Magnificence. II. — By Our Own Visitor. The gala performance at the Covent Garden Theatre last night, on the occasion of the "State Visit to the Opera, by command of her Most Gracious Majesty the Queen, in honour of his Majesty the Shah, K.G.," presented a spectacle of altogether unprecedented brilliancy. Nothing even approaching it has ever before been seen in England, and we must go back to the days of the French Empire to find a similar entertainment that could in any sense be compared with it for splendour. I arrived somewhat early and found that the ordinary drive under the portico of the theatre had been transformed into a fine and spacious vestibule [Col. 2c-3a] by reason of much bunting — above, around, and under foot — while the gas chandeliers were decorated with flowers. Entering the ordinary vestibule of the theatre I found myself in a charming ante-chamber, in which the military had already taken up their position, and in which a great number of officers, military, naval, civil, and diplomatic, were promenading. Then on through the passages into the body of the theatre, and a wonderful spectacle presented itself. The house was alive with flowers — magnificent bouquets, huge and tasteful nosegays of roses for the ladies and buttonholes for the gentlemen, rested between every alternate stall, and traced the outline of the rows of boxes on every tier in splendid profusion, and imparted a semblance of fairy-land to the interior. Of the large bouquets it was said that no fewer than seven hundred had been provided by Messrs. Wells and Segar, and that the supply in the market round the corner was suffering severely through the contract. On the back of each seat was pinned a gorgeous programme printed in blue upon a sheet of white satin twelve inches by eighteen. And now the fortunate holders of seats began to pour in — all that was most beautiful, most aristocratic, and most wealthy in London. But it was noticeable that very, very few of the well known faces in artistic, literary, or scientific circles were present. I have seen many an important function and ceremonial, but never have I seen so many beautiful women together, or such a wonderful display of diamonds and jewels. "I should like to have the chance," whispered a Bond-street gentleman in front of me to his friend, after the Princess had arrived, "I should like to have the chance of buying all the stones in this house for a million sterling. I'd put up with the balance." He was apparently "in the trade." Never has the Princess of Wales looked more beautiful than last night, as, in her deep red dress, her neck smothered in necklaces and rivières of diamonds, and a dazzling diamond coronet on her head — by the side of which all other tiaras and jewels seemed to pale — she stood in front of the box, biting her lips with what appeared to be suppressed amusement. The Royal party, all in full uniform, sat thus:— </blockquote> {| class="wikitable" |+ !Princess Victoria of Wales !Prince Henry of Battenberg !Princess Maud of Wales !Prince Albert Victor !Princess of Wales !S<small>HAH</small> !Prince of Wales !Princess Beatrice !Prince George of Wales !Princess Louise of Wales !Earl of Fife (apparently: he was behind a column) |} <blockquote> During the last interval, the Guest [the Shah] did not wait for the Princess, but left his seat and she quietly followed him out. I also left the body of the theatre, and, making my way to the vestibule, where the band of the Royal Artillery was playing and a strong guard of the same regiment was mounted, I thought I had never beheld a more picturesque or dazzling a scene. Military men in their uniforms, Life Guards in their clanking steel, Hussars with their clicking spurs, Doctors of Law in their gowns, men in Court dress, Highlanders, Volunteer officers, officers of regiments of the line, gorgeous in lace and brave in buttons, all hemmed in by a hollow square of the heavily gold-laced Queen's servants, formed a scene of brilliancy, tempered by the presence of ladies. Such was the gala night of the Covent Garden Theatre, and if the sight of it had not impressed itself for good on the heart of the Shah as he drove away in the magnificent gold carriage (for all the world like that in which the Prince and Princess drive round the ring at the Christmas circus), he must at least have felt grateful for the most superb society reunion that has ever taken place in London since London began.<ref>"The Shah at the Opera." ''Pall Mall Gazette'' 3 July 1889, Wednesday: 4 [of 8], Col. 2a–3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18890703/012/0004. Same print title and p.</ref> </blockquote> ===27 July 1889, Saturday=== Wedding between Princess Louise of Wales and the Duke of Fife at Buckingham Palace (http://www.london-gazette.co.uk/issues/25962/pages/4311/page.pdf). ==== Garden Party Hosted by Mr. and Mrs. Augustus Harris ==== The Mr. Charles Dickens who attended cannot be the novelist, but he could be his son, Charles Dickens, Jr., who died in 1896.<ref>{{Cite journal|date=2024-12-24|title=Charles Dickens Jr.|url=https://en.wikipedia.org/wiki/Charles_Dickens_Jr.|journal=Wikipedia|language=en}}</ref> Who Lord and Lady Greville are requires confirmation, as [[Social Victorians/People/Warwick|Earl and Countess Warwick]] might have been called Lord and Lady Greville; calling [[Social Victorians/People/Greville|Algernon Fulke]] and [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] Lord and Lady would be incorrect but not a big stretch. The Dowager Duchess of Newcastle in 1889 was Henrietta, 6th [[Social Victorians/People/Newcastle|Duchess of Newcastle]], who married Thomas Theobald Hohler in 1880.<blockquote>Mr. and Mrs. Augustus Harris gave a very successful garden party on Saturday, 27th ult., at The Elms, Avenue-road, when the beautiful grounds were filled with a brilliant and fashionable assemblage. The approach to the house is through a long avenue of elms, and once inside the gates one might be far away in the depths of the country. Both Mr. and Mrs. Harris stood near the entrance to welcome the guests; Mrs, Harris wearing a beautiful toilette of eau de nil silk, with a becoming French hat, with long streamers at the back. Tents were erected on the lawns, and little tables were dotted about the grass, where the guests might discuss the huge peaches at their ease. The band of the Scots Guards discoursed sweet music on the lawn, and the extremely beautiful toilettes worn by the ladies enhanced the charm of the scene. The beautiful Lady de Grey came in on her way from the Royal wedding, looking lovely in an exquisite toilette of soft white mousseline de soie, with a pointed vest of gold passementerie, and a becoming little bonnet of the same material; she wore very fine diamond earrings, which paled beside the brilliancy of her eyes. Mrs. Baskcomb had also come on from the wedding, and looked charming in a dress of white and blue mousseline de laine, with a vest and sash of dark blue silk, and a bonnet made entirely of cornflowers. Mrs. Baskcomb wore the favour from the Royal wedding (a pretty spray of orange-blossom and white heather); and round her neck was a Royal gift, consisting of a pendant with the Prince of Wales's feathers on a raised gold ground. Mrs. Phillips also wore an exquisite toilette which had figured on the same occasion. The [[Social Victorians/People/Newcastle|Dowager Duchess of Newcastle]] wore an artistic costume of bronze and green. Mrs. Heywood attracted a good deal of attention in a bright red silk dress, with a sunshade to match, having a spray of pink may at the top, and a diamond collar necklace. Mrs. Sam Lewls looked particularly striking in a green dress slightly figured with black, a grass-green mantle trimmed with bands of black passementerie, and a pretty bonnet made entirely of grass with a coronet of yellow buttercups. Mrs. Alfred Caldicott looked very elegant in a successful combination of bronze and green, with a bonnet of salmon-pink roses. Mrs. Henry Wylde looked well in white silk striped with black velvet, and a large hat covered with white ostrich feathers; and Mrs. Philip Falk was in sapphire-blue silk and velvet. A very ''chic'' toilette was worn by Mrs. Skirrow, who wore a very pretty green hat with a turned-up brim, trimmed with Japanese chrysanthemums in mauve, pink, and cream. Mrs. Maxse looked very pretty in silver-grey silk, with a long black lace mantle with winged sleeves, and a black lace hat wreathed with pink and crimson roses. The most ''récherché'' toilette of the occasion was worn by Mrs. Randegger, who looked very pretty in salmon-pink silk shot with fawn, with a bonnet of roses of the same colour. Nearly all Mr. Harris's ''prima donnas'' were present, and each formed a separate centre of attraction. Mdme. Ella Russell wore a charming Empire dress of eau de nil crêpe de chine with a coral-pink sash, & bonnet of coral-pink roses, and a diamond star at the throat. The sunshade which accompanied this fresh and dainty toilette was of white silk edged with real Brussels lace. Mdme. Valda looked very handsome in a dress of ivory silk covered with Italian lace, with a cuirass of gold passementerie, and a white tulle bonnet trimmed with gold-coloured velvet. Mdme. Marie Roze wore a fancy striped mousseline de laine, with a straw hat lined with sky-blue silk and trimmed with pink roses, the écru lace vest crossed by the large diamond stars belonging to the tiara presented to her by the people of Liverpool. Pretty Mdme. Nordica wore a Directoire dress in white-and-fawn striped silk, and a white bonnet trimmed with amber. Mrs. Bernard Beere looked very picturesque in pearl white brocade with a white lace front, a white chip hat trimmed with Boulanger carnations, a nosegay of the same in the bodice, and one of the flowers tucked into the throat under the high collar. Buttons of star sapphires were the only touch of colour in the dress. Celebrities of every description were to be seen among the guests. Mons. Max O'Rell discoursed art with Mons. Jean de Paléologue and Mons. Raphael. Mons. Ed. Lantéri was also present (professor of sculpture at South Kensington, and the designer of the medals which Mr. Harris has lately presented to his artists in commemoration of his successful operatic season) and Mons. Bertrand, who has taught the Duchess of Fife to fence, and speaks greatly of her proficiency in the graceful art. A great many of Mr. Harris's colleagues on the County Council were present, amongst whom may be mentioned Colonel Edis, Captain Probyn, Mr. E. Boulnois, M.P., Colonel Rotton, Mr. Beechcroft, Mr. Beaufoy, Mr. J. F. Beck, Mr. Carter, Mr. Carr-Gomm, Mr. Frank Debenham, Mr. Lye, Mr. A. Bassett-Hopkins, Mr. Lemon, Mr. Torr, Mr. Rhodes, Captain Verney, Mr. Walter Wren, and Mr. F. A. Ford. The rain luckily kept off until quite the end of the party, when the remaining guests adjourned to the house and examined the beautiful silver salver and marble gilt photograph stand which was presented to the popular manager last week by the principal members of the Royal Italian Opera Company, whose portraits, with autographs attached, give additional value to the graceful gift. Among the guests were:— The [[Social Victorians/People/Newcastle|Dowager Duchess of Newcastle and Mr. Hohler]], the Earl and Lady Cairns, Lord de I'lsle, Lord and Lady Hindlip, Sir Edmund and Lady Lechmere, Sir Arthur Birch, Lord Hay, '''Lord and Lady Greville''', Sir Charles and Lady Tupper, Lord and LadyHobhouse, Lady Fanny Fitzwygram, Sir Albert Rollit, M.P., Lord and Lady Raincliffe, Mr. John Aird, M.P., and Mrs. Aird, Sir Richard and Lady Wood, Mr. H. De Stern, Sir Francis and Lady Truscott, Mr. and Mrs. Ellicott, Mr. Hume Spry; Sir F. Perkins, Lady Colin Campbell, Sir Thomas and Lady Gabriel, Mr. Henry Tracy, Lady and Miss Pauncefote, Colonel Sir Robert Harley, Sir Edward Hamley, Sir Vincent and Lady Barrington, Sir W. and Lady Hardman, Mr. and Lady Agnes Cooper, Sir James and Lady Linton, Mr. and Mrs. Kendal, Sir Oscar Clayton, Mr. J. F. Firth, M.P., and Mrs. Firth, Sir F. and Lady Alston, Mdme. Roze and Col. Mapleson, Mr. T. Thorne, Alderman and Mrs. Savory, Rev. Dr. and Mrs. Kynaston, Mr. and Mrs. Arthur Stirling, Mr. and Mrs. L. Brough, Mr. and Mrs. W. P. Frith, Mr. Seymour Lucas, Mr. Woodall, M.P., Sir George Grove, Mr. L. Jennings, M.P., Col. E. Hughes, M.P., and Mrs. Hughes, Dr. W. H. Russell, Mr. and Mrs. Sydney Hoffnung[,] Mr. Pope, Q.C, Miss Bury, Dr. and Mrs. Robson Roose, Mr. H. Poland, Q.C., Sir S and Lady Gabriel, Mr. C. Whitmore, M.P., Alderman and Lady Isaacs, Sir T. and Lady Lucas, Sir R. Fowler, M.P., and Miss Fowler, Alderman Sir John and Lady Ellis, Lady Monckton, Mr. and Mrs. J. C. Parkinson, '''Mr. and Mrs. Charles Dickens''', Mr. and Mrs. Harry Furniss, Mr. G. A. Sala, Mr. and Mrs. Pinero, Mdme. Valia, Alderman and Mrs. Evans, Mr. and Mrs. A. J. Caldicott, Mr. Harry Marks, Mons. Castelmary, Mons. and Mdme. Paul Blouet, Mdme. Fursch-Madi, Mr. and Mr. [sic] Charles Wyndham, Col. R. W. Edis, Alderman and Mrs. Tyler, Mr. E. Boulnois, M.P., and Mrs. Boulnois, Mr. and Mrs. Terris, Mr. A. M. Broadley, Mr. J. Fernandez, Mr. Phil Morris, A R.A., Alderman, Mrs., and the Misses Faudel Philips, Sir W. and Miss Charley, Mr. Bidney Brough, Le Comte Ostrorog, Captain and Mr. [sic] Cdfford [Clifford?] Probyn, Mr. Under-Sheriff Clarence Halse and Mrs. Halse, Mr. and Mrs. Edward Sassoon, Le Comte and Comtesse Dillon, Mr. and Mrs. Folk, Captain Henshaw Russell, Mrs. Adair, [[Social Victorians/People/Bourke|Mr. and Mrs. Algernon Bourke]], Mr. H. V. Higgins, Captain and Mrs. Lathom Fox, Colonel and Mrs. Rotton, Signor Mancinelli, Rev. Henry White, Captain Chetwynd, Mr. and Mrs. Hume Webster, Mons. Gennadino, Alderman and Mrs. Renals, Major-General Hales Wortham, Signor and Mdme. Arditi, Rev. Canon and Mrs. Barker, Mr. and Mrs. Frank Debenham, Mrs. Gabrielli, Dr. and Mrs. Scott Gatty.<ref>"In Society." ''Lady's Pictorial'' 03 August 1889, Saturday: 62 [of 82], Col. 3a–c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18890803/425/0062. Same print title, p. 174.</ref></blockquote> === 1889 July 31, Wednesday === ==== Fete of the Uxbridge Habitation of the Primrose League ==== <blockquote>A fête, held under the auspices of the Uxbridge Habitation of the Primrose League, took place on Wednesday afternoon last in the beautiful grounds of Hillingdon Court, kindly lent by the Hon. Algernon Mills, and the evert may be described as in every way a success. A more charming spot for holding a fête of this kind could scarely be selected, and the weather was brilliant. It transpired that the arrangements for the afternoon's programme had been in a great measure suggested and carried out by Lady Mary Mills, and this being the case it was satisfactory for the large company who assembled at the meeting that was called together towards the close of the proceedings, to hear graceful compliments paid to Lady Mary, not only by Mr. C. Roberts, the Deputy Ruling Counciller of the Habitation, but by Lord Claud Hamilton, for the kind and active interest she had evinced in the event. Other necessary arrangements were very ably superintended by officers of the Habitation, aided by the Hon. Secretary, Mr. B. T. Gales. The company present included the Hon. Algernon Mills, Ruling Councillor of the Habitation, Lady Mary Mills, Lord Claud Hamilton, Lord Chesterfield, the Countess Stradbroke, [[Social Victorians/People/Bourke|the Hon. Algernon and Mrs. Bourke]], Sir Charles and Lady Cox, Mr. and Mrs. Wemyss, Mr. H. R. Cos, Mr. F. H. Deane and the Misses Deane, Mr. and Mrs. B. H. W. Way and Miss Way, General and the Misses Swanston, General Stanton, Mrs. B. C. Stephenson and party, Miss Clarke (Swakeleys), Mrs. C Grenfell, Mr. and Miss Roberts, Mr. and Mrs. J. C. Hibbert, Mrs. Wakefield, Dr. and Mrs. Ferris, Mr. H. Bird, Mr. W. Avery Bird, Mr. and Mrs. S. S. Phillips and party, the Misses Duncombe, Mr. and the Misses Grimsdale, Mr. W. B. Harman [Herman?], Mr. G. Woolls, the Rev. J. H. Thomas, T. W. James, and T. M Everett. The amusements provided during the afternoon included selections of music by members of the Uxbridge and Hillingdon Band, a very interesting entertainment of performing dogs, under the direction of Mr. Hotime, pleasing vocal and instrumental performances by the Meier Family, and several selections of comic melodies by experienced artistes, the latter of whom appeared upon a prettily decorated stage. The piano, by the way, which was used in the accompaniments, was supplied by Mrs. Willis, of Uxbridge. Tea and other refreshments were provided, and the majority of the company availed themselves of the opportunity of partaking of the refreshing beverage. Satislaction was expressed on all sides with the general success of the occasion, which certainly passed off in a manner which could not fail to please everyone present. Shortly after five o'clock the company assembled in front of a platform or stage to hear some brief addresses on political subjects. The chair was taken by the Hon. Algernon Mills, who was accompanied by Lord Claud Hamilton, M.P., and Mr. C. Roberts, and, before the meeting closed, he was joined by Mr. F. D. Dixon-Hartland, M.P., whose arrival was greeted with cheers. Lord Claud Hamilton, who was briefly introduced by the Chairman, congratulated the Habitation, first upon the successful appearance of the meeting, and next upon the great addition to their ranks which had occurred within the past few days. He thought that although Uxbridge in past days was not distinguished for its Conservatism, there had been a marked change for the better during the past few years, and he thought that when the time once more arrived the voters of Uxbridge would again be found polling in the right direction. At the same time that be congratulated the people of that district upon what he found to be their efficient local organisation, he thought they might, as a party, congratulate themselves on the present position of the Unionist Government in Parliament. This was the close of the third Session of the present Government, and although there had been one or two bye-elections, on personal and other grounds, and they had lost a few seats, yet no impartial man could deny that at the present moment the Government of Lord Salisbury was far stronger in Parliament and in the estimation of the country at large than it had been at any time during the past three years, and there was throughout the Kingdom a firm and solemn determination on the part of most electors to support end to cement the union between the Conservative party and the section of the Liberal party called Liberal Unionists. While such a feeling existed he thought that the present Government might certainly be regarded as strong. He must confess, however, that he felt sorry on taking up the morning paper that day to see that an old and personal friend, and one whom he admired, had been once again coming forward somewhat in the nature of a disturber of the public peace. He had read two speeches in the paper that day, one by Mr. A. Balfour, the other by the great politician, Lord Randolph Churchill, and in regard to the latter he was at first really unable to define what he was driving at in the remarks he made, and the policy which he attempted to unfold to his Birmingham audience, for he did not commence with a straight hit, but by side-hitting and numerous other modes of attack endeavoured to make out to his hearers that nearly everything connected with the foreign policy of Lord Salisbury was wrong, and that a great deal of the policy of Mr. Balfour in Ireland was also wrong. Having reviewed and criticised Lord Randolph's remarks in reference to foreign policy, he proceeded to deal with his observations having reference to the action of the Government in regard to Ireland, and, in this matter, confessed that he was always sorry when he heard an old friend talk undignified nonsense on a public platform, for although his lordship said a good deal that was wise and sensible, he showed his total ignorance of the present circumstances existing in regard to Ireland, and, above all, his ignorance of the Irish character. Then Lord Randolph Churchill also appeared to feel that he ought to have seat in Birmingham. Now, his hearers were doubtless very well acquainted with an interesting suburban town called Brentford. He did not think that a town like Brentford was large enough for two kings, and Birmingham was not large enough for two kings; and if two men of the calibre of Mr. Joseph Chamberlain and Lord Randolph Churchill both occupied seats for Birmingham, he did not think that it could be done without some trouble occurring in the future. After some further remarks on general political policy, his lordship concluded by once more congratulating the local Habitation of the Primrose League on the success of that day's proceedings. Mr. Dixon-Hartland, M.P., who was received with much applause, remarked, at the outset, having reference to the observations he had heard made by Lord Claud Hamilton, that he regretted Lord Randolph Churchill had spoken as he had done during the last few days. He was one or those who had watched Lord Randolph Churchill's career with great interest for a long time, and he felt that the Conservative party owed much to him. In 1880, when the fortunes of the party seemed at a low ebb, a force and courage was given to it by the way in which Lord Randolph Churchill attacked the Radical ministry that were then in power. He did good work for the Party, and that way one of the causes why Lord Salisbury was in power at the present time. However, much as he (the speaker) admired his lordship's courage and valuable work in the past, that did not blind him to what the noble lord was doing now, and the doctrines which he had recently been preaching at Walsall and at Birmingham were not Tory but socialistic doctrines. The question of doing away with rents wherever they existed, without compensation, was one that would never be accepted by the Tory party, and if Lord Randolph Churchill was going to pin his faith to a flag of that kind he would have to join the other side, for the Conservatives would not consent to it. The speaker then remarked upon the progress of the measure in reference to the Royal Grants. He remarked that last week there were few people whose hearts were not moved with the idea of the marriage of the daughter of our future king. There could be no doubt that the marriage was a thoroughly popular one with the English people, for it reminded us of the olden times, when our Kings came forward and gave their daughters in marriage to English noblemen; and the majority of people in England to-day thought our noblemen were quite as good as German princelings. (Loud applause.) English people were glad to see a return to the old state of things, for it was a custom which created another tie between the people and the Throne of England. (Hear, hear.) In Parliament most unworthy, ungenerous, and he might say, improper means had been taken to try and divide the vote for sums of money necessary to keep up the dignity of the Royal Family. For his own part he did not think that if England were polled the people of this country would be so mean as not to keep up the dignity of their Royal Family. It was a question which concerned the country more than was generally contemplated, for, in his opinion, anything that we paid to the Queen and for properly maintaining the Royal dignity was small, considering the great benefits that we received. The benefits he referred to had been bestowed upon the English people by the very constitutional way in which the queen had performed the duties of her high office, and had that not been the case we in England would not have been in the state of prosperity which might now be ascribed to us as a country, and that would have affected every man, woman and child he then saw before hm. We were told we were being unduly taxed, but, after all, what was it that was paid for the Royal Family. He submitted that at the present moment the whole of the sum annually paid amounted to rather less than one penny per head of the whole population, and it would be somewhere about one penny if the proposed grant were made. We were better off in this respect than the people of America, who paid a vast amount of money annually to Congressmen and Senators — a great deal more than we paid our queen, while, it must be remembered, the services of members of the House of Lords and House of Commons were gratuitous. We must look at the details of these things, view the question in a general, and at the same time in a common sense, way, and it would be seen that in these matters we were more lightly taxed than any country in Europe or any couuntry in the New World. Referring to the past Session of Parliament, the speaker said it had been remarkable for the breaking-up, so to speak, of the power of Mr. Gladstone over the Radical portion of his supporters; and in this respect it might be mentioned that Mr. Gladstone had come forward to say he would do what was right in regard to the royal grants. Then there was another point that might be referred to, and that was the breaking up of the Irish party, who not only felt that Mr. Bal- '''[Col. 1c–2a]''' four's government was enticing the ground from under their feet, but were also becoming aware of the fact that money was not coming in in the way it used to do. Mr. Diller had been making a tour in Australian colonies; and the general conclusion to be arrived at was that all these showed how strong and good was the government of Lord Salisbury. (Hear, hear.) The speaker then proceeded to remark that while the effect of Lord Salisbury's government was manifestly beneficial, there were apart from the influence of our own administration, signs of trouble among foreign nations, and that all this should cause the fact to be more firmly impressed upon the English mind that we should have a strong and stable government, and it behoved all in this country to do everything in their power to support the same. In conclusion, he wished to say that he was very pleased they had had such a pleasant afternoon, and he was sure they must one and all feel indebted to the Hon. Algernon Mills for the use of his grounds on the occasion. (Hear, hear.) He must remember, however, that on such occasions as that they did not assemble for the sole purpose of hearing speeches, and certainly not long speeches, and he would therefore conclude his remarks, but before doing so he wished to say that he hoped that year by year Primrose gatherings would be held in the neighbourhood of Uxbridge, and that that Habitation, and others, would continue to prosper, for there was no doubt that such societies brought all classes together, they were cemented in one bond, and where that was the case there was very little chance of the Radicals gaining any power. There was yet one other matter which he wished very briefly to refer to before he resumed his seat. He had been told lately that there had been a report in circulation to the effect that he had some thought of relinquishing his seat in Parliament as a representative of that Division. All he could say was that he considered it the greatest honour to represent a Division in the county of Middlesex, and that be was quite content with the Division he had the great honour to represent at the present time. So long as they in that Division were willing to have him, he was most willing to stick to them, and he hoped that he might be their member for many years to come, and have the pleasure of continuing to come amongst them, knowing that he was in the midst of friends. (Loud and continued cheering.) Mr. C. Roberts, who was greeted with every manifestation of popularity, then came forward for the purpose of proposing a vote of thanks to the Hon. Algernon Mills and Lady Mary Mills for their kindness in allowing the fête to take place in the beautiful grounds of Hillingdon Court. It was evident that they had the good of the cause at heart, and the fact of such support brought a great many recruits under the Primrose League standard. It was satisfactory for him to be able to state that during a very short period they had received the names of 100 recruits. Such rapid strides as that would not be made but for gatherings of that kind, which benefitted the Primrose League movement very much throughout the country; and it was undoubtedly productive of a great deal of good when the gentry opened their parks in this way, as had been the case during late years, for it brought all classes of society together, and refuted the argument that the Tories were composed of the "swells" of society, and would have nothing whatever to do with those who occupied a more humble position in life. (Hear, hear.) Such a statement as that was, of course, a great error, and they of the Primrose League, whether they belonged to the upper or the middle class, knew full well that so long as people conducted themselves the League was open to all who were of that political opinion, and that whenever there was a fête of that kind there was no distinction of persons whatever, all were perfectly free to attend the meeting, and there was certainly no idea of setting class against class. He could not help remarking that the Primrose League in that district had a most able representative in Lady Mary Mills. The whole of the entertainments on that occasion had been organised by her, and, the event having taken place in such fine weather, and those present having been so extremely well entertained, the entire proceedings might be fittingly described as a great success, and he thought it would for a fitting close to such an enjoyable occasion if a hearty vote of thanks were passed to the Hon. Algernon MiIls and Lady Mary Mills for their kindness in entertaining such a large company that afternoon. (Applause.) He did not think that such a proposal needed a seconder, and therefore he called upon all who approved of it to signify the same in the usual way. Prolonged applause was the answer to this invitation for an expression of opinion, and one enthusiastic visitor called for cheers for Lady Mary Mills, which were promptly given. The Hon. Algernon Mills replied in appropriate terms, and remarked that whatever gratification had been derived from the proceedings of that afternoon was due to the arrangements which had been entirely carried out by his wife, for he, owing to having been engaged in business matters, had been unable to devote much attention to it. All he could say was that he hoped it would not be the last meeting of the kind that would take place there. (Loud applause.)<ref>"Uxbridge Habitation of the Primrose League. Fete at Hillingdon Court." ''Uxbridge & W. Drayton Gazette'' 3 August 1889, Saturday: 8 [of 8], Col. 1a–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002284/18890803/126/0008. Print: ''Buckinghamshire Advertiser, Uxbridge, Harrow and Watford Journal'', p. 8.</ref> </blockquote> ==August 1889== ===6 August 1889, Tuesday=== The official Savoy Hotel opening:<blockquote>The Savoy Hotel on the Thames Embankment opened by a company, 6 August 1889; the directors included the earl of Lathom, Mr. R. D'Oyly Carte, and sir Arthur Sullivan.<ref>''Hayden's Dictionary of Dates and Universal Information Relating to All Ages''. Ed., Benjamin Vincent. 23rd Edition, Containing the History of the World to the End of 1903. New York: G. P. Putnam's Sons, 1904. Page 1105. ''Google Books'', retrieved 23 February 2010.</ref></blockquote>This official opening followed 2 soft openings, one a "Housewarming Supper"<ref name=":2">Williams, Olivia. ''The Secret Life of the Savoy: Glamour and Intrigue at the World's Most Famous Hotel''. Pegasus Books, 2021. Google Books (retrieved July 2025) https://www.google.com/books/edition/The_Secret_Life_of_the_Savoy/a_IDEAAAQBAJ.</ref>{{rp|1871}} hosted by the Hwfa Williamses, and the second a "private viewing."<ref name=":2" />{{rp|1872}} "Guests" at the post-opera supper included people like<blockquote>the Duchess of Manchester Consuelo Montagu, Lord Dunraven, Lord and Lady Gosford, Lord and Lady Dudley, Lord and Lady de Grey, Lord Hartington, Lord Hardwick, and the Honourable Algernon Bourke, the owner of White's who had just lost the best chef in London to D'Oyly's hotel kitchen. ... Representing the businessmen whom D'Oyly was so keen to attract were the English branch of the Rothschilds, and the American Forbes family.<ref name=":2" />{{rp|1871}}</blockquote> ===15 August 1889, Thursday=== [[Social Victorians/Pleydell-Bouverie Skelmersdale Wedding|Pleydell-Bouverie Skelmersdale Wedding]] ===26 August 1889, Monday=== Summer Bank Holiday. ===30 August 1889, Friday=== J. M. Stoddart, from Lippincott's, Arthur Conan Doyle, Oscar Wilde, and Gill met for dinner. Stoddart was looking for novels.<blockquote>The American house of Lippincott's, based in Philadelphia, had sent one of their men, J.M. Stoddart, over to England to search out some new talent. Lippincott's published a monthly magazine in both countries, and their policy was to include a complete story in each number. Stoddart called upon James Payn, editor of The Cornhill, and Payn recommended him to try Dr. Conan Doyle of Southsea. The result was the celebrated dinner for four at the Langham Hotel, Portland Place with Stoddart, Doyle, Oscar Wilde, and an Irish MP called Gill. ... Dr Doyle has left on record his impressions of Wilde" (Stavert 148).</blockquote>For Lippincott's Doyle wrote ''A Study in Scarlet'', which has "a character patterned after Oscar Wilde, and the Langdon Hotel gets a mention" (Stavert 149). Doyle's ''A Study in Scarlet'' was published in the February 1890 number, and Wilde's ''The Picture of Dorian Gray'' in the July 1890 one. ===30 August 1889, Friday=== W. A. Ayton and Anne Ayton were initiated into the Inner Order of the Golden Dawn (Gilbert 86 140). ==September 1889== ==October 1889== In October 1889, Jack the Ripper's signature and some vandalism in the form of drawings and commentary shows up in the guest book for the Lizard Hotel, Lizard Point, Cornwall (Cornwell 283). ===31 October 1889, Thursday=== Halloween. ==November 1889== ===5 November 1889, Tuesday=== Guy Fawkes Day ===13 November 1889, Wednesday=== According to the ''Morning Post'' (2 January 1890), there was a disturbance during a performance of ''The Gold Craze'' at the Royal Princess's Theatre:<blockquote>The Marquis de Leuville appeared yesterday at Marlborough-street Police-court, to answer a summons obtained by Mr. H. Cummings for inciting persons to create a riot at the Royal Princess's Theatre on November 13. Mr. Geoghegan stated the case for the complainant, and examined Thomas Vincent Kiely, who said that the Marquis had given him money to engage men to hiss and make a disturbance at the performance of "The Gold Craze" at the Princess's, in which the Baron de Fleurville was upposed by defendant to be [col. 5/6] intended for himself. The hearing was adjourned.<ref>"." ''Morning Post'' 2 January 1890 (No. 36,676), Thursday: p. 4, Cols. 5–6.</ref></blockquote> ==December 1889== ===7 December 1889, Saturday=== Gilbert and Sullivan's ''The Gondoliers, Or the King of Barataria'' opened at the Savoy. ===17 December 1889, Tuesday=== [[Dangan-Neville Wedding|The Wedding of Lord Dangan and Lady Violet Nevill]] ===25 December 1889, Wednesday=== Christmas Day ===26 December 1889, Thursday=== Boxing Day Opening of the Horniman Museum<blockquote>Mr. F. J. Horniman's museum at Forest-Hill was open to the public on Thursday, Friday, and Saturday in Christmas week, and about 4500 persons of all classes availed themselves of the privilege of inspecting the various objects of interest and the magnificent collection of natural history and art specimens.<ref name=":1">''Illustrated London News'' (London, England), Saturday, January 04, 1890; pg. 6; Issue 2646, Col. 2.</ref></blockquote> ===27 December 1889, Friday=== Horniman Museum open, second day.<ref name=":1" /> ===28 December 1889, Saturday=== Horniman Museum open, third day.<ref name=":1" /> ===31 December 1889, Tuesday===rib1119/8090**** ==== Robert Browning's funeral in Westminster Abbey ==== Bret Harte attended.<ref>Nissen, Axel. ''Bret Harte: Prince and Pauper''. Jackson, MS: University Press of Mississippi, 2000: 216.</ref><blockquote>The mortal remains of Robert Browning, the great poet, were yesterday laid among the relics of other famous bards and writers in Poets' Corner, Westminster Abbey, with every appropriate demonstration of honour and regret for the departed. Among the attendant mourners were many of the great of the land and a host of others, whose reverence for the poet was shown by their demeanour.<ref>"Funeral of Robert Browning." The ''Morning Post'' 1 January 1890 (No. 36,675): 4, Col. 6. ''British Newspaper Archive'' .</ref></blockquote>From later in the ''Morning Post'':<blockquote>With manifestations of sincere respect and fond remembrance the remains of Robert Browning were yesterday consigned to an honoured resting place in Westminster Abbey. The funeral procession, which included 10 mourning carriages, left the house in De Vere-gardens, Kensington, soon after eleven o'clock, and reached Westminster Abbey just after the hour of noon had struck. Long before that time the choir and transepts reserved for mourners and friends were full, with the exception of a few seats under the Lantern set apart for the relatives and pall-bearers. The north transcept and a portion of the nave were crowded with the general public. Gloom and mist pervaded the Abbey, rendering the use of artificial light imperative. Tall tapers were burning upon the footpace before the undraped altar, and the seven-branched candelabra at the entrance of the sacrarium shed a flickering light within. In the uncertain light it was difficult to discern and identify the occupants of the choir and transepts, but now and again some well-known representative of art and letters could be recognised passing to his allotted seat. Just before noon the Dean, Canons Prothero, Duckworth, Westcott, and Fruse, Minor Canons Troutbeck and Cheadle, and the choir assembled at the western entrance of the nave in readiness to meet the body. Some 10 minutes afterwards the solemn and familiar strains of Croft and Purcell's music indicated that the Burial Service had commenced, and, with slow and measured steps the long procession passed up the nave into the choir, where choir and clergy took their places in the stalls. The coffin, covered with a violet-coloured pall, upon which were placed two floral wreaths and a cross of violets, was placed upon trestles in the open space under the Lantern where the lectern usually stands. The pall-bearers, ranged on either side, were the Hon. Hallam Tennyson, Dr. Butler, Sir Fitzjames Stephen, Sir Theodore Martin, Archdeacon Farrar, Professor Masson, Professor Jowett, Sir Frederick Leighton, Sir James Paget, Sir George Grove, Mr. G. M. Smith, and Professor Knight. [new paragraph] When the mourners had taken their places, the Burian Service proceeded in the usual manner, the 90th Psalm being chanted to Purcell's setting, while the Dean read the Lesson from his stall. The anthem or hymn which is usually interpolated between the two sections of the service was yesterday preceded by a "Meditation," in which some touching verses by Elizabeth Barrett Browning were set to equally touching music by Dr. Bridge. Rendered by the choir with excellent skill and reverent feeling, this interlude formed the most striking feature in the whole service. The anthem which followed was Wesley's, "All go to one place; all are of the dust, and all turn to dust again." At its conclusion the procession reformed, and the coffin, preceded by clergy and choir, passed through the south transept to the familiar Poets' Corner, where, at the foot of Chaucer's tomb, the grave had been dug. On the wall above, suspended from Cowley's monument, was a noble wreath of laurel, bound by a broad ribbon of white silk bearing the name of the poet in golden lettering, the memorial tribute of the Municipality of Venice. The final prayers at the graveside were said by the Dean, and the service fitly closed with Dr. Watts's hymn, "O God, our help in ages past," to the time-honoured tune, "St. Anne," which was very generally taken up by the congregation. As the Dean was reading the touching prayer of committal, the rays of the winter sun glinted through the Te Deum window in the transept, shedding rainbow hues on wall and pillar, and partially dismissing the prevailing gloom. Then the strains of the Dead March pealed forth from the organ, clergy and choir retired, and friends pressed forward to take a farewell glance. [new paragraph] Among them were noticed the Archbishop of Canterbury, the Earl of Pembroke, the Earl of Rosebery, Bishop Barry, the Dean of Windsor, Sir Arthur Sullivan, Sir Lyon Playfair, M.P., Captain Walter Campbell (representing the Queen), Mr. Stuart-Wortley, M.P., Mr. Mundella, M.P., Mr. Leonard Courtney, M.P., the Hon. and Rev. E. Carr Glyn, Dr. Martineau, Canon Benham, Mr. Alma Tadema, Mr. Lecky, Mr. Oscar Wilde, and Mr. Huxley (representing his father). The relatives and immediate friends were Mr. and Mrs. R. Browning, Mr. Octavius Moulton Barrett, Mr. George Moulton Barrett, Captain Charles Moulton Barrett, Mr. Henry Moulton Barrett, M. Dourlane, Miss Christine Browning, Captain Altham, Rev. Mr. Altham, Sir James Carmichael, and Miss Carmichael. Wreaths of exquisite beauty and in almost countless numbers were sent from friends far and near, and many of them bore touching inscriptions, as for instance, that of Sir F. Leighton, "In remembrance of 36 years' friendship." Among others may be mentioned Lord Tennyson, Earl and Countess Brownlow, Lord Vernon, Lord and Lady Edmond Fitzmaurice, Lady Lindsay, Lady Martin, Sir Henry Thompson, Sir John Millais, Miss Cobbe, Mrs. Jeune, Mr. and Mrs. Alma Tadema, Mr. and Mrs. Whistler, Mr. and Mrs. Cyril Flower, Mr. Henry Irving, Mr. and Mrs. Bancroft, and the Browning Society. The coffin, made in Venice, was of yellow pine, designed somewhat in the modern casket form, and bore the simple inscription, "Robert Browning, born May 7, 1812, died December 12, 1889."<ref>"Funeral of Robert Browning." The ''Morning Post'' 1 January 1890 (No. 36,675): 5, Col. 6. ''British Newspaper Archive'' .</ref></blockquote>Professor Rücker's lecture on electricity:<blockquote>At the Royal Institution, Albemarle-street, yesterday afternoon, Professor Rücker delivered the second of his course of six simplified lectures on electricity. The theatre was again well filled. Before taking up the threads of his subject at the point where they were broken off last Saturday, the lecturer briefly recapitulated the remarks made on that occasion relative to the subject of the "lines of force."<ref name=":0">"Lectures on Electricity." The ''Morning Post'' 1 January 1890 (No. 36,675): 6, Col. 3. British Newspaper Archive .</ref></blockquote>The rest of the report describes the experiment used to demonstrate the answers to particular questions about electricity. The report ends with this sentence: "The third lecture takes place to-morrow," i.e., 1 January 1890.<ref name=":0" /> ==Works Cited== *Bernstein, Susan David. "Radical Readers at the British Museum: Eleanor Marx, Clementina Black, Amy Levy." Nineteenth-Century Gender Studies 3.2 (Summer 2007): http://ncgsjournal.com/issue32/bernstein.htm. {{reflist}} kegfj2fn0pe62nxuoqr760dieziffid 0 266023 2834854 2834840 2026-09-28T13:47:25Z ThaniosAkro 2805358 /* Solving M by (M+1) */ 2834854 wikitext text/x-wiki ==Introduction== The word "simultaneous" implies that two or more equations exist in the same context at the same time. For example: <math>3x + 2y - 13 = 0\ \dots\ (1)</math> <math>2x - y - 4 = 0 \ \dots\ (2)</math> Equations <math>(1)</math> and <math>(2)</math> are 2 simultaneous equations. When we see two simultaneous equations, it's natural to ask the question: What are the values of <math>x</math> and <math>y</math> that satisfy both equations <math>(1)</math> and <math>(2)</math>? To calculate <math>(x,y)</math> common to both <math>(1)</math> and <math>(2)</math>: <math>(2)*2\ \dots\ 4x - 2y - 8 = 0 \ \dots\ (3)</math> <math>(3)+(1)\ \dots\ 7x - 21 = 0;\ x = 3.</math> Substitute <math>3</math> for <math>x</math> in <math>(2):\ 2(3) - y - 4 = 0;\ y = 2.</math> The values <math>(x,y)=(3,2)</math> satisfy both equations <math>(1)</math> and <math>(2).</math> In cartesian coordinate geometry of 2 dimensions, point <math>(3,2)</math> is the point of intersection of two lines <math>(1)</math> and <math>(2).</math> See Figure 1 below. If the 2 lines are parallel, there is no point of intersection. See Figure 2 below. The trivial solution <math>(0, 0)</math> is a valid solution. See Figure 3 below. {{RoundBoxTop|theme=2}} <gallery> File:0726_2lines.png|<small>Figure 1. 2 lines that intersect at point (3,2).</small> File:0726_2parallelLines.png|<small>Figure 2. When 2 lines are parallel, there is no point of intersection.</small> File:0729_2lines.png|<small>Figure 3. The trivial solution x=y=0 is a valid solution.</small> </gallery> {{RoundBoxBottom}} The information contained in equations <math>(1)</math> and <math>(2)</math> may constitute a matrix: <pre> [ [3, 2, -13], [2, -1, -4] ] </pre> In fact, in the python programming language, the following statement is valid code: <syntaxhighlight lang=python> data = [ [3, 2, -13], [2, -1, -4] ] </syntaxhighlight> For there to be a point of intersection, <code>data</code> must be valid. Here are examples of invalid data: <syntaxhighlight lang=python> data = [ [3, 2, -13], # Both lines are parallel. [6, 4, -5] # See Figure 2 above. ] </syntaxhighlight> <syntaxhighlight lang=python> data = [ [3, 2, -13], [0, 0, -4] # Empty row, representing equation 0x + 0y - 4 = 0. ] </syntaxhighlight> <syntaxhighlight lang=python> data = [ [3, 0, -13], # Empty column representing 2 parallel lines: [2, 0, -4] # 3x = 13 and 2x = 4. ] </syntaxhighlight> The matrix above with name "data" is described as matrix "2 by 3," meaning that it contains 2 rows with 3 columns or 3 members per row. ==Solving 2 by 3== ===Implementation=== Below is python code containing function <code>solve2by3 (input).</code> The function accepts as input a matrix like <code>data</code> above. After the code are examples of invocation with both good data and bad data. <syntaxhighlight lang=python> import sys import decimal dD = decimal.Decimal def solve2by3 (input_, level=0) : ''' input represents system: a1*x + b1*y + c1 = 0 a2*x + b2*y + c2 = 0 input = ( (a1,b1,c1), (a2,b2,c2), ) To invoke: x,y = solve2by3 (input) Function solve2by3 () may return None. input is not changed by this function. ''' if level : thisName = 'solve2by3 (input_, level = {}) :'.format(level) else : thisName = 'solve2by3 (input_) :' try : error = 0 try : input = [ list(v) for v in input_ ] (a1,b1,c1), (a2,b2,c2) = input except : input = 0 input or {}['input_ not recognized.'] # six_values = a1,b1,c1, a2,b2,c2 for v in six_values : isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.'] set1 = { isinstance(v, dD) for v in six_values } if len(set1) == 2 : # Mix of Decimal and non-Decimal in input. Make all type Decimal. (a1,b1,c1, a2,b2,c2) = [ dD(str(v)) for v in six_values ] # a1 or b1 or {}['Row 1 empty.'] ; a1 or a2 or {}['Column 1 empty.'] a2 or b2 or {}['Row 2 empty.'] ; b1 or b2 or {}['Column 2 empty.'] # if not a1 : # Make a1 non-zero. (a1,b1,c1),(a2,b2,c2) = (a2,b2,c2), (a1,b1,c1) # if a2 : # Make a2 zero. if a2 == a1 : a2,b2,c2 = a1-a2, b1-b2, c1-c2 elif a2 == -a1 : a2,b2,c2 = a1+a2, b1+b2, c1+c2 else : list1 = [ a2*v for v in (a1,b1,c1) ] list2 = [ a1*v for v in (a2,b2,c2) ] a2,b2,c2 = [ (v1-v2) for v1,v2 in zip (list1, list2) ] a2 and {}['Internal error. a2 should be 0.'] b2 or {}['Parallel input.'] # The matrix is: # a1, b1, c1 with a1 non-zero # 0, b2, c2 and b2 non-zero. # 0 x + b2 y + c2 = 0 y = -c2/b2 # a1 x + b1 y + c1 = 0 x = -(b1*y + c1)/a1 # Here is why a1 must be non-zero. output = x,y except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ] # if error : print (thisName) for v in error : print (' ', v.strip()) if input : print (' input of solve2by3() =') print (' ', input[0]) print (' ', input[1]) return None # return output </syntaxhighlight> ====Examples==== Invocation with good data: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve2by3([ [3, 2, -13], [2, -1, -4] ]) print (output) </syntaxhighlight> <pre> (3.0, 2.0) </pre> {{RoundBoxBottom}} Invocation with bad or invalid data: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve2by3([ [3, 2, -13], # Both lines are parallel. [6, 4, -5] # See Figure 2 above. ]) print (output) </syntaxhighlight> <pre> solve2by3 (input_) : <class 'KeyError'> KeyError('Parallel input.') input of solve2by3() = [3, 2, -13] [6, 4, -5] None </pre> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve2by3([ [3, 2, -13, 7], # Input matrix not exactly 2 by 3. [6, 4, -5] ]) print (output) </syntaxhighlight> <pre> solve2by3 (input_) : <class 'KeyError'> KeyError('input_ not recognized.') None </pre> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve2by3([ [3, 0, 2], [6, 0, 4] ]) print (output) </syntaxhighlight> <pre> solve2by3 (input_) : <class 'KeyError'> KeyError('Column 2 empty.') input of solve2by3() = [3, 0, 2] [6, 0, 4] None </pre> {{RoundBoxBottom}} ===Reversing the process=== ====Linear Function==== [[File:1212line01.png|thumb|400px|''' Figure 1. Diagram of line defined by 2 points.''' </br> 2 points are: <math>(-5,4), (3,1).</math> </br> Equation of line is: <math>3x + 8y - 17 = 0.</math> ]] Examples above calculate the point of intersection of 2 lines. An example of reversing the process is this task: Given two points <math>(x_1, y_1), (x_2, y_2)</math> calculate equation of line containing these 2 points. Equation of line is of format <math>Ax + By + C = 0,</math> For example: <math>3x + 2y + 10 = 0\ \dots\ (1).</math> Equation <math>(1)</math> can be expressed as <math>6x + 4y + 20 = 0</math> or <math>-9x - 6y - 30 = 0.</math> Also, there appear to be 3 unknowns, but only two sets of values are required to define the line. Assume <math>C = 1</math> and the 2 relevant equations are: <math>Ax_1 + By_1 + 1 = 0</math> <math>Ax_2 + By_2 + 1 = 0</math> To solve for <math>A, B:</math> <math>x_1A + y_1B + 1 = 0</math> <math>x_2A + y_2B + 1 = 0</math> Suppose the line is defined by points <math>(3,1), (-5,4):</math> <syntaxhighlight lang=python> # python code: input = [ [3,1,1], [-5,4,1], ] result = solve2by3 (input) print (result) </syntaxhighlight> <syntaxhighlight> (-0.17647058823529413, -0.47058823529411764) </syntaxhighlight> Desired equation is: <math>-0.17647058823529413\cdot x -0.47058823529411764\cdot y + 1 = 0\ \dots\ (2).</math> To improve appearance of <math>(2):</math> <syntaxhighlight lang=python> # python code: print (A/B) </syntaxhighlight> <syntaxhighlight> 0.37500000000000006 </syntaxhighlight> Let <math>A, B = 3, 8</math> and check: <syntaxhighlight lang=python> # python code A,B = 3,8 p1 = x1,y1 = 3,1 p2 = x2,y2 = -5,4 for v in (p1,p2) : x,y = v C = -(A*x + B*y) s1 = 'C' print (s1, eval(s1)) </syntaxhighlight> <syntaxhighlight> C -17 C -17 </syntaxhighlight> Desired equation <math>(2)</math> becomes <math>3x + 8y - 17 = 0\ \dots\ (3).</math> ====Wrong assumptions==== [[File:1212line02.png|thumb|400px|''' Figure 2. Diagram of line defined by 2 points and passing through origin.''' </br> 2 points are: <math>(3,2), (6,4).</math> </br> Equation of line is: <math>2x - 3y = 0.</math> </br> Non-zero value for coefficient <math>C</math> is impossible. ]] Sometimes you create a matrix that seems correct, but it fails to produce a result. Here is such an example: What is equation of line defined by points <math>(3, 2), (6, 4)?</math> As above assume that <math>C = 1.</math> <syntaxhighlight lang=python> # python code: p1 = x1,y1 = 3,2 p2 = x2,y2 = 6,4 input = [ [x1,y1,1], [x2,y2,1], ] result = solve2by3 (input) print (result) </syntaxhighlight> This assumption failed: <syntaxhighlight> solve2by3() : parallel input. input = [3, 2, 1] [6, 4, 1] None </syntaxhighlight> Assume that <math>A = 1.</math> <syntaxhighlight lang=python> # python code: # 1(x1) + B(y1) + 1(C) = 0 # 1(x2) + B(y2) + 1(c) = 0 # y1(B) + 1(C) + x1 = 0 # y2(B) + 1(C) + x2 = 0 input = [ [y1,1,x1], # Notice that input has been [y2,1,x2], # rotated left by one column. ] result = solve2by3 (input) print (result) </syntaxhighlight> <syntaxhighlight> (-1.5, 0.0) </syntaxhighlight> Equation of line is: <math>1\cdot x + (-1.5)\cdot y + 0 = 0</math> or <math>2x - 3y = 0</math> This line includes the origin. Therefore constant <math>C</math> cannot be non-zero. Generally: * Without rotation, <math>A, B = \text{result}</math> and <math>C = 1.</math> * With rotation left by one column, <math>B, C = \text{result}</math> and <math>A = 1.</math> In this case: <syntaxhighlight lang=python> # python code: >>> B,C = result ; A = 1 >>> A,B,C >>> 1,-1.5,0 </syntaxhighlight> For example of big matrix that failed the first time, but was successful the second time, see [https://en.wikiversity.org/wiki/Solving_simultaneous_equations#Cubic_Resistive_Network cubic resistive network] below. ==Solving 3 by 4== Below is python code containing function <code>solve3by4 (input).</code> The function accepts as input a 3 by 4 matrix like that described in the code comments. In 3d coordinate geometry the equations <code>(1), (2), (3)</code> are the equations of 3 planes and the solution of the system is the point of intersection of the 3 planes. Given 3 random planes, there are 2 conditions that do not produce a unique point of intersection: * If 2 of the planes are parallel. See Figure 1 below. * If the 3 planes form a "tent" or "delta." See Figure 2, 3 or 4 below. A special case of this condition occurs when size of "tent" is zero, in which case all three planes intersect on a common line and there is an infinite number of solutions. See Figure 5 below. The trivial solution <math>(0, 0, 0)</math> is valid. See Figure 6 below. <code></code> <code></code> {{RoundBoxTop|theme=2}} <gallery> File:0726_2parallelPlanes.png|<small>Figure 1. When 2 planes are parallel, there is no point of intersection.</small> File:0730triangularTube.png|<small>Figure 2. Triangular tube, 3 planes that never intersect.</small> File:0726_3planes0.png|<small>Figure 3. 3 planes that form a "tent," view 1. There is no point of intersection.</small> File:0726_3planes1.png|<small>Figure 4. 3 planes that form a "tent," view 2. There is no point of intersection.</small> File:1109_3planes1.png|<small>Figure 5. 3 planes that intersect on a common line. Points <math>(6,6,1),</math> <math>(7,7,4),</math> <math>(8,8,7)</math> are on this line and are common to all 3 planes.</small> File:0729_3planes.png|<small>Figure 6. 3 planes that intersect at origin (0, 0, 0).</small> </gallery> {{RoundBoxBottom}} ===Auxiliary functions=== <syntaxhighlight lang=python> def two_planes_parallel (plane1, plane2) : ''' status = two_planes_parallel (plane1, plane2) status = True if parallel. ''' A1,B1,C1 = plane1[:3] A2,B2,C2 = plane2[:3] A3 = B1*C2 - B2*C1 B3 = C1*A2 - C2*A1 C3 = A1*B2 - A2*B1 return ({ A3,B3,C3 } == {0}) def check_three_rows (input) : ''' result = check_three_rows (input) result is None or string. ''' row0,row1,row2 = input if two_planes_parallel (row0,row1) : return 'Two rows parallel: 0,1' if two_planes_parallel (row0,row2) : return 'Two rows parallel: 0,2' if two_planes_parallel (row1,row2) : return 'Two rows parallel: 1,2' A0,B0,C0 = row0[:3] ; A1,B1,C1 = row1[:3] ; A2,B2,C2 = row2[:3] # Line at intersection of planes 1,2 has direction numbers: A3 = B1*C2 - B2*C1 ; B3 = C1*A2 - C2*A1 ; C3 = A1*B2 - A2*B1 # Line at intersection of planes 0,2 has direction numbers: A4 = B0*C2 - B2*C0 ; B4 = C0*A2 - C2*A0 ; C4 = A0*B2 - A2*B0 # Are 2 lines of intersection parallel? if two_planes_parallel ((A3,B3,C3), (A4,B4,C4)) : return 'Three rows form tent.' return None </syntaxhighlight> ===Implementation=== <syntaxhighlight lang=python> def solve3by4 (input_, level=0) : ''' input_ is : [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ] representing system: a0*x + b0*y + c0*z + d0 = 0 a1*x + b1*y + c1*z + d1 = 0 a2*x + b2*y + c2*z + d2 = 0 x,y,z = solve3by4 (input) Function solve3by4 () may return None. input is not changed by this function. ''' if level : thisName = 'solve3by4 (input_, level = {}) :'.format(level) else : thisName = 'solve3by4 (input_) :' try : error = 0 try : input = [ list(v) for v in input_ ] (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input except : input = 0 input or {}['input_ not recognized.'] # twelve_values = a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2 for v in twelve_values : isinstance(v, (int, float, dD)) or {}['Non-numeric value in input.'] set1 = { isinstance(v, dD) for v in twelve_values } if len(set1) == 2 : # Mix of Decimal and non-Decimal in input. Make all type Decimal. (a0,b0,c0,d0, a1,b1,c1,d1, a2,b2,c2,d2) = [ dD(str(v)) for v in twelve_values ] # a0 or b0 or c0 or {}['Row 0 empty.'] ; a0 or a1 or a2 or {}['Column 0 empty.'] a1 or b1 or c1 or {}['Row 1 empty.'] ; b0 or b1 or b2 or {}['Column 1 empty.'] a2 or b2 or c2 or {}['Row 2 empty.'] ; c0 or c1 or c2 or {}['Column 2 empty.'] # input = [ [a0,b0,c0,d0], [a1,b1,c1,d1], [a2,b2,c2,d2] ] result = check_three_rows (input) ; result and {}[result] # if not a0 : # Make a0 non-zero. if a1 : input[0],input[1] = input[1],input[0] else : input[0],input[2] = input[2],input[0] # # The matrix is: # a0, b0, c0, d0 with a0 non-zero. # a1, b1, c1, d1 # a2, b2, c2, d2 # Process rows so that a1, a2 become zero. row0 = input[0] ; a0 = row0[0] for p in (1,2) : rowp = input[p] ; ap = rowp[0] if ap : if ap == a0 : input[p] = [ (v1-v2) for (v1,v2) in zip(row0, rowp) ] elif ap == -a0: input[p] = [ (v1+v2) for (v1,v2) in zip(row0, rowp) ] else : input[p] = [ (v0*ap-vp*a0) for (v0,vp) in zip(row0, rowp) ] if input[p][0] : {}['Internal error. input[{}][0] should be 0.'.format(p)] (a0,b0,c0,d0), (a1,b1,c1,d1), (a2,b2,c2,d2) = input # The matrix is: # a0, b0, c0, d0 with a0 non-zero. # 0, b1, c1, d1 # 0, b2, c2, d2 two_rows = (b1,c1,d1), (b2,c2,d2) output = solve2by3 (two_rows, level-1) if output : y,z = output # a0 x + b0 y + c0 z + d0 = 0 x = -(b0*y + c0*z + d0)/a0 # Here is why a0 must be non-zero. output = x,y,z except : error = str(sys.exc_info())[1:].split(',') ; error[1:] = [ ','.join(error[1:-1]) ] # if error : print (thisName) for v in error : print (' ', v.strip()) if input : print (' input of solve3by4() =') print (' ', input[0]) print (' ', input[1]) print (' ', input[2]) return None # return output </syntaxhighlight> ====Examples==== Below are examples of invocation with both good data and bad data. Invocation with good data: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve3by4([ [2, -1, 3, -17], [1, 3, 2, -25] , [3, 2, -1, -12] , ]) print (output) </syntaxhighlight> <pre> (3.0, 4.0, 5.0) </pre> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve3by4([ [12, -12, -5, -24], [-6, 15, 8, 12] , [ 0, 10, 3, 0] , ]) print (output) </syntaxhighlight> <pre> (2.0, 0.0, -0.0) </pre> {{RoundBoxBottom}} Invocation with bad or invalid data: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve3by4([ [20, 16, 2, 17], [-77, -27, -25, -24], [81, 45, 18, 43] ]) print (output) </syntaxhighlight> <pre> solve3by4 (input_) : <class 'KeyError'> KeyError('Three rows form tent.') input of solve3by4() = [20, 16, 2, 17] [-77, -27, -25, -24] [81, 45, 18, 43] None </pre> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve3by4([ [34, -5, -21, 30], [-54, -33, 63, 62], [18, 11, -21, 38] ]) print (output) </syntaxhighlight> <pre> solve3by4 (input_) : <class 'KeyError'> KeyError('Two rows parallel: 1,2') input of solve3by4() = [34, -5, -21, 30] [-54, -33, 63, 62] [18, 11, -21, 38] None </pre> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solve3by4([ [20, 16, 2, 17], [-77, -27, '-25', -24], [81, 45, 18, 43] ]) print (output) </syntaxhighlight> <pre> solve3by4 (input_) : <class 'KeyError'> KeyError('Non-numeric value in input.') input of solve3by4() = [20, 16, 2, 17] [-77, -27, '-25', -24] [81, 45, 18, 43] None </pre> {{RoundBoxBottom}} ===Reversing the Process=== ====Quadratic function==== [[File:1212quadratic01.png|thumb|400px|''' Figure 1. Graph of quadratic function defined by 3 points.''' ]] A quadratic function <math>y = f(x) = Ax^2 + Bx + C</math> is defined by 3 points: <math>(-2, 9.0), (5, 4.8), (10, 13.8).</math> What is <math>f(x)?</math> <math>y = Ax^2 + Bx + C</math> <math>A, B, C</math> are 3 unknowns. <math>x^2(A) + x(B) + 1(C) - y = 0</math> <syntaxhighlight lang=python> # python code: x1,y1 = -2, 9 x2,y2 = 5, 4.8 x3,y3 = 10, 13.8 input = [ [x1**2, x1, 1, -y1], [x2**2, x2, 1, -y2], [x3**2, x3, 1, -y3], ] A, B, C = solve3by4 (input) s1 = 'A,B,C' print (s1, eval(s1)) </syntaxhighlight> <syntaxhighlight> A,B,C (0.2, -1.2, 5.8) </syntaxhighlight> <math>y = f(x) = 0.2x^2 - 1.2x + 5.8</math> ====Wrong assumptions==== [[File:1212plane02.png|thumb|400px|''' Figure 2. Graph of plane containing X axis.''' </br> Because plane passes through origin, non-zero value for coefficient <math>D</math> is impossible. </br> Because plane is parallel to <math>X</math> axis, non-zero value for coefficient <math>A</math> is impossible. ]] In 3 dimensions a plane (<math>Ax + By + Cz + D = 0</math>) is defined by 3 points: <math>(5, 3, -2), (-1, 12, -8), (37, 15, -10).</math> What is equation of plane? <math>Ax + By + Cz + D = 0</math> <math>xA + yB + zC + D = 0</math> Assume that <math>D = 1.</math> <math>xA + yB + zC + 1 = 0</math> <syntaxhighlight lang=python> # python code pt1 = x1,y1,z1 = (5, 3, -2) pt2 = x2,y2,z2 = (-1, 12, -8) pt3 = x3,y3,z3 = (37, 15, -10) input = [ [ 5, 3, -2, 1], [-1, 12, -8, 1], [37, 15, -10, 1], ] result = solve3by4 (input) s1 = 'result' print (s1, eval(s1)) </syntaxhighlight> <syntaxhighlight> solve2by3() : parallel input. input = (-63, 42, -6) (36, -24, 32) checkDirectionNumbers() : 3 rows form a tent. input of solve3by4() = (5, 3, -2, 1) (-1, 12, -8, 1) (37, 15, -10, 1) result None </syntaxhighlight> Input was invalid. Rotate input left and try again. <syntaxhighlight lang=python> # python code. input = [ [ 3, -2, 1, 5], [12, -8, 1, -1], [15, -10, 1, 37], ] result = solve3by4 (input) s1 = 'result' print (s1, eval(s1)) </syntaxhighlight> <syntaxhighlight> solve2by3() : empty column in input. input = (0, 9, 63) (0, 12, -36) checkDirectionNumbers() : 3 rows form a tent. input of solve3by4() = (3, -2, 1, 5) (12, -8, 1, -1) (15, -10, 1, 37) result None </syntaxhighlight> Input was invalid. Rotate input left and try again. <syntaxhighlight lang=python> input = [ [ -2, 1, 5, 3], [ -8, 1, -1, 12], [-10, 1, 37, 15], ] result = solve3by4 (input) s1 = 'result' print (s1, eval(s1)) </syntaxhighlight> <syntaxhighlight> result (1.5, 0, 0) </syntaxhighlight> Without rotation: <math>A, B, C = \text{result}</math> and <math>D = 1.</math> With rotation left: <math>B, C, D = \text{result}</math> and <math>A = 1.</math> With another rotation left: <math>C, D, A = \text{result}</math> and <math>B = 1.</math> <syntaxhighlight lang=python> # python code: C, D, A = (1.5, 0, 0) ; B = 1 A, B, C, D </syntaxhighlight> <syntaxhighlight> (0, 1, 1.5, 0) </syntaxhighlight> Equation of plane is: <math>0x + 1y + 1.5z + 0 = 0</math> or <math>2y + 3z = 0</math> =====A better method===== {{RoundBoxTop|theme=3}} There are four unknown values: <math>A,B,C,D.</math> Create matrix 4 by 5: <syntaxhighlight lang=python> # python code. pt1 = x1,y1,z1 = (5, 3, -2) pt2 = x2,y2,z2 = (-1, 12, -8) pt3 = x3,y3,z3 = (37, 15, -10) input = [] for (x_,y_,z_) in (pt1,pt2,pt3) : x,y,z = [ dD(v) for v in (x_,y_,z_) ] input += [ [ x,y,z,1,0 ] ] # For valid plane at least one of A,B,C must be non-zero. # For last condition let A + B + C = 5. input += [ [ 1,1,1,0,-5 ] ] A,B,C,D = solveMbyN(input) print (input) print ( '({})x + ({})y + ({})z + ({}) = 0'.format(A,B,C,D) ) </syntaxhighlight> <syntaxhighlight> [[ 5, 3, -2, 1, 0], # 5A + 3B - 2C + 1D + 0 = 0 [-1, 12, -8, 1, 0], # -1A + 12B - 8C + 1D + 0 = 0 [37, 15, -10, 1, 0], # 37A + 15B - 10C + 1D + 0 = 0 [ 1, 1, 1, 0, -5]] # 1A + 1B + 1C + 0D - 5 = 0 (0)x + (2)y + (3)z + (0) = 0 </syntaxhighlight> Equation of plane is: <math>2y + 3z = 0.</math> {{RoundBoxBottom}} ==Solving M by (M+1)== Below is python code containing function <code>solveMbyN (input).</code> The function accepts as input a 4 by 5 matrix (or greater) like that described in the code comments. This function is recursive, calling itself with M being reduced by 1 with each recursive invocation until the matrix is size 3 by 4 in which case function solve3by4() is called. <syntaxhighlight lang=python> # python code import decimal Decimal = decimal.Decimal getcontext = decimal.getcontext def reduceRow (row) : ''' This function calculates the max(abs) of all values in row, divides all values by the max(abs) and returns the result. [27, -15, 42] becomes [27/42, -15/42, 42/42]. Very small values are converted to zero. To invoke: output = reduceRow (row) output is list or None ''' if False in [ isinstance(v,Decimal) for v in row ] : print ('reduceRow: non Decimal in input.', { type(v) for v in row }) return None max = sorted([ abs(v) for v in row ])[-1] row = [ v/max for v in row ] almostZero = Decimal('1e-' + str( getcontext().prec )) row = [ (v, Decimal(0))[abs(v) < almostZero] for v in row ] return row enable_reduceRow = 0 def solveMbyN (input) : ''' input represents system: a0*w + b0*x + c0*y + d0*z + e0 = 0 a1*w + b1*x + c1*y + d1*z + e1 = 0 a2*w + b2*x + c2*y + d2*z + e2 = 0 a3*w + b3*x + c3*y + d3*z + e3 = 0 or greater. This solves a matrix of M rows by N columns where N = M+1 and M >= 2 input = [ (a0,b0,c0,d0,e0), (a1,b1,c1,d1,e1), (a2,b2,c2,d2,e2), (a3,b3,c3,d3,e3), ] To invoke: w,x,y,z = solveMbyN (input) Function solveMbyN () may return None. If input is size 15 by 16: v1,v2,v3, .... ,v14,v15 = solveMbyN (input) input is not changed by this function. ''' def solveMbyN_(input) : input = [ list(v) for v in input ] # input is now local, hard copy of original input. # This ensures that original input is not changed by this function. numberOfRows = len(input) numberOfColumns = numberOfRows+1 if numberOfRows < 2 : print ('solveMbyN()! len(input) < 2.') return None if numberOfRows == 2 : return solve2by3(input) if numberOfRows == 3 : return solve3by4(input) for row in range (0, numberOfRows) : if len(input[row]) != numberOfColumns : print ('solveMbyN()! row with bad length.') print (' row number',row,'of',numberOfRows,'rows has length',len(input[row]),'.') return None if True not in [ bool(v) for v in input[row][:-1] ] : print ('solveMbyN()! empty row in input, row number',row,'of',numberOfRows,'rows.') return None for column in range (0, numberOfRows) : sum = 0 for row in range (0, numberOfRows) : sum = bool(input[row][column]) if sum : break if not sum : print ('solveMbyN()! empty column in input. column number',column,'of',numberOfRows,'columns.') return None for column in (numberOfRows,) : sum = 0 for row in range (0, numberOfRows) : sum = bool(input[row][column]) if sum : break if not sum : # Empty column. Trivial solution is valid. return (0,) * numberOfRows if enable_reduceRow : # For matrices greater than 10 by 11 enable this piece of code. for row in range (0, numberOfRows) : data = reduceRow (input[row]) if isinstance(data,list) : input[row] = data # Arrange input if necessary so that a0 is non-zero. if not input[0][0] : for p in range (1, numberOfRows) : if input[p][0] : input[p], input[0] = input[0], input[p] break # The matrix is: # [a0,b0,c0,d0,e0], with a0 non-zero, # [a1,b1,c1,d1,e1], # [a2,b2,c2,d2,e2], # [a3,b3,c3,d3,e3], # Process rows so that a1, a2, a3, .... become zero. a0 = input[0][0] for p in range (1, numberOfRows) : ap = input[p][0] if ap : L0 = [ input[0][q]*ap for q in range (0, numberOfColumns) ] Lp = [ input[p][q]*a0 for q in range (0, numberOfColumns) ] L_ = [ L0[q]-Lp[q] for q in range (0, numberOfColumns) ] if L_[0] : # This should not happen. print ('solveMbyN()! internal error 1.') return None input[p] = L_ # The matrix is: # [a0, b0, c0, d0, e0], with a0 non-zero, # [ 0,b1_,c1_,d1_,e1_], # [ 0,b2_,c2_,d2_,e2_], # [ 0,b3_,c3_,d3_,e3_], data1 = [ input[p][1:] for p in range (1, numberOfRows) ] # data1 = [ # [b1_,c1_,d1_,e1_], # [b2_,c2_,d2_,e2_], # [b3_,c3_,d3_,e3_] ] data2 = solveMbyN(data1) # Here is where function solveMbyN() calls itself. if data2 == None : return None topRow = input[0][1:] # b0, c0, d0, e0 topRow # x, y, z data2 listOfProducts = [ topRow[p]*data2[p] for p in range (0, len(data2)) ] + topRow[-1:] # listOfProducts = [ b0*x, c0*y, d0*z, e0 ] sumOfProducts = [ sum for sum in (0,) for p in listOfProducts for sum in (sum+p,) ][-1] # sumOfProducts = [ b0*x, b0*x+c0*y, b0*x+c0*y+d0*z, b0*x+c0*y+d0*z+e0 ][-1] # sumOfProducts = b0*x + c0*y + d0*z + e0 # a0*w + b0*x + c0*y + d0*z + e0 = 0 # a0*w + sumOfProducts = 0 w = -sumOfProducts/a0 # Here is why a0 must be non-zero. return (w,) + data2 output = error = '' try : output = solveMbyN_(input) except : error = sys.exc_info()[:2] if (output == None) or error : if error : print ('solveMbyN()!') print (' error detected in processing.') print (' ', error) if len(input) >= 4 : print (' input of solveMbyN() with',len(input),'rows =') for row in input : print (' ', row) return None return output </syntaxhighlight> ===Examples=== Below are examples of invocation with both good data and bad data. Invocation with good data: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> L1 = [ [Decimal('-870442133162'), Decimal('239132906684'), Decimal('397660499756'), Decimal('788825405004'), Decimal('156830886652'), Decimal('-87096140169')], [Decimal('-112130420352'), Decimal('-93605595828'), Decimal('-185262083472'), Decimal('-93369903383'), Decimal('-497973505065'), Decimal('-766710012513')], [Decimal('-142983492168'), Decimal('619713461197'), Decimal('-608752759383'), Decimal('-448477443042'), Decimal('-281161661751'), Decimal('-799463977245')], [Decimal('572854919541'), Decimal('-148673693908'), Decimal('341291756040'), Decimal('362403188059'), Decimal('-202136092949'), Decimal('-429932229578')], [Decimal('157632861972'), Decimal('330790263276'), Decimal('-836474627955'), Decimal('644295395386'), Decimal('-595590967129'), Decimal('-392835045727')] ] output = solveMbyN(L1) print (output) </syntaxhighlight> The following results were produced with decimal precision set at 50: <pre> ( Decimal('-0.06331410760212760088313271882866553044211194694576'), Decimal('1.0969089506763419155148060427588009931309074487534'), Decimal('1.0837038580207682607126722016017867670831200359767'), Decimal('-0.42989566124678842791674322699930461340596671792215'), Decimal('-2.0541601084281682582146753421426610779297051535362') ) </pre> {{RoundBoxBottom}} Invocation with bad or invalid data: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solveMbyN([ [-4, -6, 3, -4, -8, -8], [1, 0, 2, 5, 1, 7], [9, -3, -8, 4, 7, -5], [1, -9, 5, -2, 5, -2], [1, -3, -6, -3, -6, 6] ]) print (output) </syntaxhighlight> <pre> # With each step towards solve2by3() the length of each number in digits can theoretically double. solve2by3() : parallel input. input = (-181440, 181440, -101088) (90720, -90720, 406944) checkDirectionNumbers() : 3 rows form a tent. input of solve3by4() = [-756, -1176, 0, -1872] [-324, -744, 240, -936] [-324, -384, -120, -264] input of solveMbyN() with 4 rows = [-6, 11, 16, -4, 20] [-66, -5, -20, -44, -92] [-42, 23, -12, 12, -16] [-18, -21, -16, -32, 16] input of solveMbyN() with 5 rows = [-4, -6, 3, -4, -8, -8] [1, 0, 2, 5, 1, 7] [9, -3, -8, 4, 7, -5] [1, -9, 5, -2, 5, -2] [1, -3, -6, -3, -6, 6] None </pre> Provided that numbers are small, integers, floats and Decimal objects produce the same results: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solveMbyN([ [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0], [1.0, 0.0, 2.0, 5.0, 1.0, 7.0], [9.0, -3.0, -8.0, 4.0, 7.0, -5.0], [1.0, -9.0, 5.0, -2.0, 5.0, -2.0], [1.0, -3.0, -6.0, -3.0, -6.0, 6.0] ]) print (output) </syntaxhighlight> <pre> solve2by3() : parallel input. input = (-181440.0, 181440.0, -101088.0) (90720.0, -90720.0, 406944.0) checkDirectionNumbers() : 3 rows form a tent. input of solve3by4() = [-756.0, -1176.0, 0.0, -1872.0] [-324.0, -744.0, 240.0, -936.0] [-324.0, -384.0, -120.0, -264.0] input of solveMbyN() with 4 rows = [-6.0, 11.0, 16.0, -4.0, 20.0] [-66.0, -5.0, -20.0, -44.0, -92.0] [-42.0, 23.0, -12.0, 12.0, -16.0] [-18.0, -21.0, -16.0, -32.0, 16.0] input of solveMbyN() with 5 rows = [-4.0, -6.0, 3.0, -4.0, -8.0, -8.0] [1.0, 0.0, 2.0, 5.0, 1.0, 7.0] [9.0, -3.0, -8.0, 4.0, 7.0, -5.0] [1.0, -9.0, 5.0, -2.0, 5.0, -2.0] [1.0, -3.0, -6.0, -3.0, -6.0, 6.0] None </pre> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> output = solveMbyN([ [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')], [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')], [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')], [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')], [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')] ]) print (output) </syntaxhighlight> <pre> solve2by3() : parallel input. input = (Decimal('-181440'), Decimal('181440'), Decimal('-101088')) (Decimal('90720'), Decimal('-90720'), Decimal('406944')) checkDirectionNumbers() : 3 rows form a tent. input of solve3by4() = [Decimal('-756'), Decimal('-1176'), Decimal('0'), Decimal('-1872')] [Decimal('-324'), Decimal('-744'), Decimal('240'), Decimal('-936')] [Decimal('-324'), Decimal('-384'), Decimal('-120'), Decimal('-264')] input of solveMbyN() with 4 rows = [Decimal('-6'), Decimal('11'), Decimal('16'), Decimal('-4'), Decimal('20')] [Decimal('-66'), Decimal('-5'), Decimal('-20'), Decimal('-44'), Decimal('-92')] [Decimal('-42'), Decimal('23'), Decimal('-12'), Decimal('12'), Decimal('-16')] [Decimal('-18'), Decimal('-21'), Decimal('-16'), Decimal('-32'), Decimal('16')] input of solveMbyN() with 5 rows = [Decimal('-4'), Decimal('-6'), Decimal('3'), Decimal('-4'), Decimal('-8'), Decimal('-8')] [Decimal('1'), Decimal('0'), Decimal('2'), Decimal('5'), Decimal('1'), Decimal('7')] [Decimal('9'), Decimal('-3'), Decimal('-8'), Decimal('4'), Decimal('7'), Decimal('-5')] [Decimal('1'), Decimal('-9'), Decimal('5'), Decimal('-2'), Decimal('5'), Decimal('-2')] [Decimal('1'), Decimal('-3'), Decimal('-6'), Decimal('-3'), Decimal('-6'), Decimal('6')] None </pre> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Sphere=== In cartesian coordinate geometry of three dimensions a sphere is represented by equation: <math>x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.</math> On the surface of a certain sphere there are 4 known points: <syntaxhighlight lang=python> # python code point1 = (13,7,20) point2 = (13,7,4) point3 = (13,-17,4) point4 = (16,4,4) </syntaxhighlight> What is equation of sphere? Rearrange equation of sphere to prepare for creation of input matrix: <math>(x)A + (y)B + (z)C + (1)D + (x^2 + y^2 + z^2) = 0.</math> Create input matrix of size 4 by 5: <syntaxhighlight lang=python> # python code input = [] for (x,y,z) in (point1, point2, point3, point4) : input += [ ( x, y, z, 1, (x**2 + y**2 + z**2) ) ] print (input) </syntaxhighlight> <syntaxhighlight> [ (13, 7, 20, 1, 618), (13, 7, 4, 1, 234), (13, -17, 4, 1, 474), (16, 4, 4, 1, 288), ] # Matrix containing 4 rows with 5 members per row. </syntaxhighlight> <syntaxhighlight lang=python> # python code result = solveMbyN(input) print (result) </syntaxhighlight> <syntaxhighlight> (-8.0, 10.0, -24.0, -104.0) </syntaxhighlight> Equation of sphere is : <math>x^2 + y^2 + z^2 - 8x + 10y - 24z - 104 = 0</math> ===Quintic function=== [[File:1212quintic01.png|thumb|400px|''' Figure 4. Graph of quintic function defined by 6 points.''' ]] A quintic function <math>y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F</math> is defined by 6 points: <math>(-8, 0), (-6, 0), (-3, 0), (1,0), (3,0), (5,10).</math> What is <math>f(x)?</math> <math>y = f(x) = Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F</math> <math>A, B, C, D, E, F</math> are 6 unknowns. <math>x^5(A) + x^4(B) + x^3(C) + x^2(D) + x(E) + 1(F) - y = 0</math> <syntaxhighlight lang=python> # python code: pt1 = x1,y1 = -8,0 pt2 = x2,y2 = -6,0 pt3 = x3,y3 = -3,0 pt4 = x4,y4 = 1,0 pt5 = x4,y5 = 3,0 pt6 = x6,y6 = 5,10 input = [] for pt in (pt1,pt2,pt3,pt4,pt5,pt6) : x,y = pt input += [[ x**5, x**4, x**3, x**2, x, 1, -y ]] print (input) </syntaxhighlight> <syntaxhighlight> [[-32768, 4096, -512, 64, -8, 1, 0], [ -7776, 1296, -216, 36, -6, 1, 0], [ -243, 81, -27, 9, -3, 1, 0], [ 1, 1, 1, 1, 1, 1, 0], [ 243, 81, 27, 9, 3, 1, 0], [ 3125, 625, 125, 25, 5, 1, -10]] </syntaxhighlight> <syntaxhighlight> result = solveMbyN (input) s1 = 'result' print (s1, eval(s1)) </syntaxhighlight> <syntaxhighlight> result (0.0010926573426573423, 0.014204545454545447, 0.027316433566433533, -0.18028846153846156, -0.3343531468531468, 0.47202797202797203) </syntaxhighlight> From these results the following ratios may be deduced: <math>\frac{1}{A} = 915.2;\ </math> <math>\frac{B}{A} = 13;\ </math> <math>\frac{C}{A} = 25;\ </math> <math>\frac{D}{A} = -165;\ </math> <math>\frac{E}{A} = -306;\ </math> <math>\frac{F}{A} = 432.</math> <math>y = f(x) = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}</math> ===Resistive network=== {{RoundBoxTop|theme=2}} [[File:1127network01.png|thumb|400px|''' Figure 1. Diagram of resistive network containing 8 resistors.''' ]] A resistive network containing 8 resistors is connected as shown in Figure 1. What is resistance <math>R</math> between points <math>P_1, P_3?</math> {{RoundBoxBottom}} ====Solve for 8 Branch Currents==== {{RoundBoxTop|theme=2}} [[File:1127network02.png|thumb|400px|''' Figure 2. Diagram of resistive network showing all branch currents.''' </br> </br> At any point to which 3 or more conductors are connected, sum of currents = 0. </br> At point <math>P_4</math> currents <math>i_1, i_7</math> flow towards point; current <math>i_4</math> flows away from point. </br> <math>i_1 + i_7 - i_4 = 0.</math> </br> </br> In any closed loop sum of voltages = 0. </br> In loop containing points <math>P_1, P_2</math> voltages <math>e_2, e_8</math> are clockwise; voltage <math>e_3</math> is counter-clockwise. <math></math> </br> <math>e_2 + e_8 - e_3 = 0.</math><math></math><math></math> </br> </br> Direction of flow through <math>r_7, r_8</math> is assumed and is subject to correction. ]] Voltage across <math>r_1:</math> Using <math>E = IR,\ e_1 = i_1\cdot r_1 = r_1\cdot a.</math> Similarly: <math>e_2 = r_2\cdot b.</math> <math>e_3 = r_3\cdot c.</math> <math>\dots\dots</math> <math>e_8 = r_8\cdot h.</math> Build a matrix size 8 by 9 which will be input to function solveMbyN(). <syntaxhighlight lang=python> # Python code: r1,r2,r3,r4,r5,r6,r7,r8 = 10,20,30,40,50,60,70,80 input = [] # e1 = e2 + e7 # 10a = 20b + 70g # 10a - 20b + 0c + 0d + 0e + 0f - 70g + 0h + 0 = 0 input += [[r1, - r2, 0, 0, 0, 0, - r7, 0, 0]] # e3 = e2 + e8 # 30c = 20b + 80h # 0a - 20b + 30c + 0d + 0e + 0f + 0g - 80h + 0 = 0 input += [[0, - r2, r3, 0, 0, 0, 0, - r8, 0]] # e5 = e7 + e4 # 50e = 70g + 40d # 0a + 0b + 0c - 40d + 50e + 0f - 70g + 0h + 0 = 0 input += [[0, 0, 0, - r4, r5, 0, - r7, 0, 0]] # e5 = e8 + e6 # 50e = 80h + 60f # 0a + 0b + 0c + 0d + 50e - 60f + 0g - 80h + 0 = 0 input += [[0, 0, 0, 0, r5, - r6, 0, - r8, 0]] # i7 + i5 + i8 = i2 # g + e + h = b # 0a - 1b + 0c + 0d + 1e + 0f + 1g + 1h + 0 = 0 input += [[0 ,- 1, 0, 0, 1, 0, 1, 1, 0]] # i4 = i1 + i7 # d = a + g # -1a + 0b + 0c + 1d + 0e + 0f - 1g + 0h + 0 = 0 input += [[-1, 0, 0, 1, 0, 0, - 1, 0, 0]] # i6 = i3 + i8 # f = c + h # 0a + 0b - 1c + 0d + 0e + 1f + 0g - 1h + 0 = 0 input += [[0, 0, - 1, 0, 0, 1, 0, - 1, 0]] # Total input current = 1 # a + b + c = 1 # 1a + 1b + 1c + 0d + 0e + 0f + 0g + 0h - 1 = 0 input += [[1, 1, 1, 0, 0, 0, 0, 0, - 1]] </syntaxhighlight> This last entry can contain almost any arbitrary values. For example: <syntaxhighlight lang=python> # Python code: # i7 = 2.3 input += [[0, 0, 0, 0, 0, 0, 1, 0, - 2.3]] # e5 = 1.9 input += [[0, 0, 0, 0, r5, 0, 0, 0, - 1.9]] </syntaxhighlight> Changing this last entry changes all values of <math>e_x, i_x.</math> However, the ratio <math>\frac{e_x}{i_x} = r_x</math> remains constant. {{RoundBoxTop|theme=3}} input is matrix of size 8 by 9: <syntaxhighlight> [ [10, -20, 0, 0, 0, 0, -70, 0, 0], # e1 = e2 + e7, or r1(i1) - r2(i2) - r7(i7) = 0 [ 0, -20, 30, 0, 0, 0, 0, -80, 0], # e3 = e2 + e8, or r3(i3) - r2(i2) - r8(i8) = 0 [ 0, 0, 0, -40, 50, 0, -70, 0, 0], # e5 = e4 + e7, or r5(i5) - r4(i4) - r7(i7) = 0 [ 0, 0, 0, 0, 50, -60, 0, -80, 0], # e5 = e6 + e8, or r5(i5) - r6(i6) - r8(i8) = 0 [ 0, -1, 0, 0, 1, 0, 1, 1, 0], # i2 = i5 + i7 + i8, or 1(i5) + 1(i7) + 1(i8) - 1(i2) = 0 [-1, 0, 0, 1, 0, 0, -1, 0, 0], # i4 = i1 + i7, or 1(i4) - 1(i1) - 1(i7) = 0 [ 0, 0, -1, 0, 0, 1, 0, -1, 0], # i6 = i3 + i8, or 1(i6) - 1(i3) - 1(i8) = 0 [ 1, 1, 1, 0, 0, 0, 0, 0, -1] # i1 + i2 + i3 = 1, or 1(i1) + 1(i2) + 1(i3) - 1 = 0 ] </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> i1,i2,i3,i4,i5,i6,i7,i8 = values_of_i = solveMbyN (input) s1 = 'values_of_i' ; print(s1,eval(s1)) s1 = 'i1 + i2 + i3' ; print(s1,eval(s1)) </syntaxhighlight> <syntaxhighlight> values_of_i (0.45727389222364606, 0.3065353746543468, 0.23619073312200717, 0.4350171983543536, 0.31685438726647336, 0.24812841437917316, -0.022256693869292507, 0.011937681257165982) i1 + i2 + i3 1.0 </syntaxhighlight> <math>i_7</math> has negative value. Reverse <math>i_7</math> to make all branch currents positive. {{RoundBoxBottom}} ====Make all Branch Currents positive==== {{RoundBoxTop|theme=2}} [[File:1127network03.png|thumb|400px|''' Figure 3. Diagram of resistive network showing all branch currents positive.''' ]] Make <math>i_7</math> positive and check results. <syntaxhighlight lang=python> # Python code i7 *= -1 values_of_e = e1,e2,e3,e4,e5,e6,e7,e8 = [ eval(s1) for s in '12345678' for s1 in [ 'i' + s + '*r' + s ] ] s1 = 'values_of_e' print(s1,eval(s1)) </syntaxhighlight> <syntaxhighlight> values_of_e [ 4.572738922236461, 6.130707493086937, 7.085721993660215, 17.400687934174144, 15.842719363323669, 14.88770486275039, 1.5579685708504756, 0.9550145005732785] </syntaxhighlight> <syntaxhighlight lang=python> # python code t1 = ( 'e1+e7-e2', # e1 + e7 should equal e2 'e2+e8-e3', # e2 + e8 should equal e3 'e7+e5-e4', # e7 + e5 should equal e4 'e8+e6-e5', # e8 + e6 should equal e5 'e1+e4-(e2+e5)', # e1 + e4 should equal e2 + e5 'e1+e4-(e3+e6)', # e1 + e4 should equal e3 + e6 'e1+e7+e8-e3', # e1 + e7 + e8 should equal e3 'e7+e8+e6-e4', # e7 + e8 + e6 should equal e4 'i1-i7-i4', # i1 should equal i7 + i4 'i6-i3-i8', # i6 should equal i3 + i8 'i2+i7-(i8+i5)', # i2 + i7 should equal i8 + i5 'i1+i2+i3 - (i4+i5+i6)', # Total input current should # equal total output current. ) for s1 in t1 : print(s1,eval(s1)) </syntaxhighlight> <syntaxhighlight> e1+e7-e2 -8.881784197001252e-16 e2+e8-e3 0.0 e7+e5-e4 0.0 e8+e6-e5 0.0 e1+e4-(e2+e5) 0.0 e1+e4-(e3+e6) 0.0 e1+e7+e8-e3 -8.881784197001252e-16 e7+e8+e6-e4 0.0 i1-i7-i4 -5.551115123125783e-17 i6-i3-i8 6.938893903907228e-18 i2+i7-(i8+i5) 0.0 i1+i2+i3 - (i4+i5+i6) 0.0 </syntaxhighlight> {{RoundBoxBottom}} ====Compute R==== {{RoundBoxTop|theme=8}} <syntaxhighlight lang=python> # Python code E1 = e3+e6 I1 = i1 + i2 + i3 R1 = E1/I1 s1 = 'R1' ; print(s1,eval(s1)) E2 = e1 + e7 + e5 I2 = i4 + i5 + i6 R2 = E2/I2 s1 = 'R2' ; print(s1,eval(s1)) </syntaxhighlight> <syntaxhighlight> R1 21.973426856410605 R2 21.973426856410605 </syntaxhighlight> <math>R = R_1 = R_2</math> {{RoundBoxBottom}} ==Solving big matrices== A big matrix is one of size greater than 10 by 11. With each recursive call of solveMbyN() the size of each value in decimal digits can theoretically double. Python's integer math enjoys infinite precision. If your original matrix is of size 32 by 33, and each value in the matrix is an int of 1 decimal digit, function solve2by3() can be called with ints of size 2**30 (1,073,741,824) decimal digits. Floats can be used instead of ints, but the limited precision of floats may not provide a desirable level of accuracy in the result. Python's decimal objects are a good compromise and function reduceRow() above is an attempt to keep the size of each member manageable. Function reduceRow() works well. However, as with all operations involving python's decimal objects, you need to pay careful attention to precision and accuracy of the results. On my Mac the following python code generates a matrix of size 100 by 101 and calculates the result containing 100 values in about 1.5 seconds, including the time to generate 10,100 random numbers, each containing up to 12 decimal digits with the decimal point in a random position, and including the time to test output over 100 rows to verify that all calculated values of the solution satisfy each row. In function solveMbyN() above find line: <syntaxhighlight lang=python> if 0 : # For matrices greater than 10 by 11 enable this piece of code. </syntaxhighlight> Change it to: <syntaxhighlight lang=python> if 1 : # For matrices greater than 10 by 11 enable this piece of code. </syntaxhighlight> Then: <syntaxhighlight lang=python> from random import getrandbits from decimal import * import sys getcontext().prec=50 # Set precision to 50. numRows=100 def randomNumber(): # 1 invocation in 256 returns 0. if not (getrandbits(10) & 0xFF) : return Decimal(0) return [ w for q in [ getrandbits(10) & 1] for sign in [ (-1,1)[q] ] for numberOfDecimalPlaces in [ getrandbits(10) & 0xF ] for s1 in [ str(getrandbits(50))[-12:] ] for s2 in [ '0'*12+s1 ] for posn in [ len(s2)-numberOfDecimalPlaces ] for w in [ sign*Decimal(s2[:posn]+'.'+s2[posn:]) ] ][0] data = [[ randomNumber() for p in range(numRows+1) ] for p in range (numRows) ] datan = sorted([ v for row in data for v in row if v < 0 ]) datap = sorted([ v for row in data for v in row if v >= 0 ]) print ('A small sample of the random numbers in data:') print ('+ve values:',len(datap), ',', datap[:2],'\n ',datap[-2:]) print ('-ve values:',len(datan), ',', datan[:2],'\n ',datan[-2:]) output = solveMbyN(data) print ('A small sample of the solutions:') print ('output[0] =', output[0]) print ('output[19] =', output[19]) print ('output[59] =', output[59]) print ('output[99] =', output[99]) </syntaxhighlight> <pre> A small sample of the random numbers in data: +ve values: 5021 , [Decimal('1.16871265E-7'), Decimal('1.73721245E-7')] [Decimal('998523464710'), Decimal('999768605679')] -ve values: 5079 , [Decimal('-991079944209'), Decimal('-989786083110')] [Decimal('-4.10272824E-7'), Decimal('-2.38664537E-7')] A small sample of the solutions: output[0] = 0.43689400018920695253771832432651757885163362928745 output[19] = 0.19867480862268284151047885891372314311902973127033 output[59] = 0.46169681824826309421940885550155275946668017604646 output[99] = -0.020598220802233733529683040085023443421113627790464 </pre> ===Testing results=== Data input to function solveMbyN() is an array of 100 rows, each row containing 101 numbers. <pre> [ [v0_0, v0_1, v0_2, ..... v0_98,v0_99,v0_100], ....... [v99_0,v99_1,v99_2, ...... v99_98,v99_99,v99_100],] ] </pre> Data received from function solveMbyN() is an array of 100 numbers. <pre> [ s0,s1,s2, ...... s98,s99 ] </pre> The first row is tested with the solutions by; <pre> L2 = [ v0_0*s0, v0_1*s1, v0_2*s2, ..... v0_98*s98,v0_99*s99,v0_100 ] </pre> sum is the sum of all items in L2. sum is appended to L1 and the process is repeated for all 100 rows. <syntaxhighlight lang=python> L1 = [] for row in data : # a, b, c, d, e, f, g ....... # row # u, v, w, x, y, z # output L2 = [ row[p]*t1[p] for t1 in (output+(1,),) for p in range (0, len(row)) ] # L2 = [a*u , b*v , c*w , d*x , e*y , f*z , g] sum = [ sum for sum in [0] for p in L2 for sum in [sum+p] ][-1] # sum = [a*u, a*u + b*v, a*u + b*v + c*w, a*u + b*v + c*w + d*x, a*u + b*v + c*w + d*x + e*y, # a*u + b*v + c*w + d*x + e*y + f*z, a*u + b*v + c*w + d*x + e*y + f*z + g][-1] # sum = a*u + b*v + c*w + d*x + e*y + f*z + g # sum is the sum of all items in L2 L1 += [abs(sum)] L1a = sorted(L1) print (len(L1a), L1a[0], L1a[-1]) </syntaxhighlight> Ideally sum should be zero. In practice it's extremely unlikely that sum will be zero. My results typically gave values close to <pre> 100 4.1065149609E-38 9.6425614873452398368906E-25 </pre> The worst of the results above was 23 zeroes after the decimal point. Better results can be achieved by increasing precision. ===Cubic Resistive Network=== {{RoundBoxTop|theme=2}} [[File:1129network01.png|thumb|400px|''' Figure 1. Diagram of resistive network containing 12 resistors.''' </br> Blue arrows show assumed direction of current. ]] A resistive network containing 12 resistors is connected as shown in Figure 1. The network is called "cubic" because, in 3 dimensions, each resistor is located on one edge of a cube. What is resistance <math>R</math> between points <math>P_1, P_2?</math> <syntaxhighlight lang=python> # Python code: values_of_r = r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12 = 10,20,30,40,50,60,70,80,90,100,110,120 </syntaxhighlight> {{RoundBoxBottom}} ====Solve for 12 Branch Currents==== =====Create matrix 12 by 13===== <syntaxhighlight lang=python> # Python code: input = [ [10, 20, -30, -40, 0, 0, 0, 0, 0, 0, 0, 0, 0], # e1 + e2 = e3 + e4 [ 0, 0, 0, 0, 50, 60, -70, -80, 0, 0, 0, 0, 0], # e5 + e6 = e7 + e8 [ 0, 0, 0, 40, 0, 0, 0, -80, 0, 0, 110, -120, 0], # e11 + e4 = e8 + e12 [10, 0, 0, 0, -50, 0, 0, 0, 90, -100, 0, 0, 0], # e9 + e1 = e5 + e10 [ 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0], # i5 = i10 + i6 [ 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0], # i7 = i11 + i8 [-1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], # i9 = i1 + i3 [ 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], # i4 = i3 + i11 [-1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], # i2 = i1 + i10 [ 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0], # i12 = i6 + i8 [ 0, -1, 0, -1, 1, 0, 1, 0, 1, 0, 0, -1, 0], # i9 + i5 + i7 = i2 + i4 + i12 [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1.6] # i3 = 1.6 ] i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input) s1 = 'values_of_i' print(s1,eval(s1)) </syntaxhighlight> This matrix failed: <syntaxhighlight> solve2by3() : empty row in input. input = (0, 0, 0) (90533116137774317568000000000000000000000000000000000000000000, -84449318308603494400000000000000000000000000000000000000000000, 1.4228848407261846e+61) checkDirectionNumbers() : 2 rows parallel, 1 and 2. input of solve3by4() = [38760268800000000000000000000, 593753233920000000000000000000, -569385344000000000000000000000, 0] [620164300800000000000000000000000000, 9500051742720000000000000000000000000, -9110165504000000000000000000000000000, 0] [153839206400000000000000000000000, 20882470400000000000000000000000, -81126144000000000000000000000000, -3.67098806272e+32] </syntaxhighlight> =====Try again===== <syntaxhighlight lang=python> # Python code: input = [ [10, 20, -30, -40, 0, 0, 0, 0, 0, 0, 0, 0, 0], # e1 + e2 = e3 + e4 [ 0, 0, 0, 0, 50, 60, -70, -80, 0, 0, 0, 0, 0], # e5 + e6 = e7 + e8 [ 0, 0, 0, 40, 0, 0, 0, -80, 0, 0, 110, -120, 0], # e11 + e4 = e8 + e12 [10, 0, 0, 0, -50, 0, 0, 0, 90, -100, 0, 0, 0], # e9 + e1 = e5 + e10 [ 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0], # i5 = i10 + i6 [ 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0], # i7 = i11 + i8 [-1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], # i9 = i1 + i3 [ 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], # i4 = i3 + i11 [-1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], # i2 = i1 + i10 [ 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0], # i12 = i6 + i8 #[ 0, -1, 0, -1, 1, 0, 1, 0, 1, 0, 0, -1, 0], # i9 + i5 + i7 = i2 + i4 + i12 DELETED. [10, 20, 0, 0, 0, 0, -70, -80, 90, 0, 0, -120, 0], # e9 + e1 + e2 = e7 + e8 + e12 ADDED. [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1.6] # i3 = 1.6 ] import decimal dD = decimal.Decimal dgt = decimal.getcontext() Precision = dgt.prec = 30 Tolerance = dD("1e-" + str(Precision-2)) # Convert all values in input to decimal objects with precision of 30. input = [ [ dD(str(p)) for p in v ] for v in input ] i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12 = values_of_i = solveMbyN (input) </syntaxhighlight> <syntaxhighlight> # values_of_i 5.5027171492204899777282850777 9.3809275973899113038721525414 1.5999999999999999999999999999 # i3 should be 1.6. 4.8661430860000781463681475402 6.12901340210213730316883522816 2.25080295393271597702496776458 4.68526862814050716992927753670 1.41912554214042902356112999643 7.10271714922048997772828507766 3.87821044816942132614386746358 3.26614308600007814636814754027 3.66992849607314500058609776103 </syntaxhighlight> ====Compute R==== {{RoundBoxTop|theme=8}} <syntaxhighlight lang=python> # python code values_of_e = e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12 = [ (i*r) for (i,r) in zip(values_of_i, values_of_r) ] </syntaxhighlight> <syntaxhighlight> values_of_e 55.0271714922048997772828507770 187.618551947798226077443050828 47.9999999999999999999999999970 194.645723440003125854725901608 306.450670105106865158441761408 135.048177235962958621498065875 327.968803969835501895049427569 113.530043371234321884890399714 639.244543429844097995545656989 387.821044816942132614386746358 359.275739460008596100496229430 440.391419528777400070331731324 </syntaxhighlight> <math>e_3</math> should be <math>48,\ i_3*r_3</math> or <math>1.6*30.</math> <syntaxhighlight lang=python> E1 = e5 + e10 + e2 I1 = i9 + i5 + i7 R1 = E1/I1 s1 = 'R1' ; print(s1,eval(s1)) E2 = e7 + e8 - e6 + e10 - e1 + e3 + e4 I2 = i2 + i4 + i12 R2 = E2/I2 s1 = 'R2' ; print(s1,eval(s1)) </syntaxhighlight> <syntaxhighlight> R1 49.2208688540148837936367036806 R2 49.2208688540148837936367036803 </syntaxhighlight> Display results with precision of 28: <syntaxhighlight lang=python> dgt.prec = 28 R1 += 0 ; R2 += 0 dgt.prec = Precision s1 = 'R1' ; print(s1,eval(s1)) s1 = 'R2' ; print(s1,eval(s1)) </syntaxhighlight> <syntaxhighlight> R1 49.22086885401488379363670368 R2 49.22086885401488379363670368 </syntaxhighlight> <math>R = R_1 = R_2.</math> {{RoundBoxBottom}} ====Review==== In matrix all values <math>i_1, i_2, \dots, i_{11}, i_{12}</math> and <math>e_1, e_2, \dots, e_{11}, e_{12}</math> are included. Testing was done with the following lines substituted for last line of matrix: <syntaxhighlight lang=python> # python code [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -3.15] # i7 = 3.15 [ 0, 0, 0, r4, 0, 0, 0, 0, 0, 0, 0, 0, -6 ] # e4 = 6 [ 0, 0, r3, 0, 0, 0, 0, r8, 0, 0, 0, 0, -5 ] # e3 + e8 = 5 </syntaxhighlight> All tests produced a consistent value of <math>R.</math> ====Checking==== =====Examples of current nodes===== {{RoundBoxTop|theme=2}} <gallery> File:0101network01.png|<small><math>i_1 + i_{10} = i_2</math></small> File:0101network02.png|<small><math>i_6 + i_8 = i_{12}</math></small> </gallery> {{RoundBoxBottom}} =====Examples of voltage loops===== {{RoundBoxTop|theme=2}} <gallery> File:0108network00.png|<small>Top and Bottom.<math>e_1 + e_2 - e_4 - e_3 = 0</math><math>e_5 + e_6 - e_8 - e_7 = 0</math></small> File:0108network01.png|<small>Two sides in background.<math>e_5 + e_{10} - e_1 - e_9 = 0</math><math>e_6 + e_{12} - e_2 - e_{10} = 0</math></small> File:0108network02.png|<small>Two sides in foreground.<math>e_7 + e_{11} - e_3 - e_9 = 0</math><math>e_8 + e_{12} - e_4 - e_{11} = 0</math></small> </gallery> {{RoundBoxBottom}} =====Python code===== <syntaxhighlight lang=python> t1 = ( # 3 nodes on top 'i1 + i10 - i2', 'i1 + i3 - i9', 'i3 + i11 - i4', # 3 nodes on bottom 'i10 + i6 - i5', 'i11 + i8 - i7', 'i6 + i8 - i12', # Total input current equals total output current. 'i5 + i7 + i9 - i2 - i4 - i12', # Voltage loops # top 'e1 + e2 - e4 - e3', # bottom 'e5 + e6 - e8 - e7', # 2 sides in background 'e9 + e1 - e10 - e5', 'e10 + e2 - e12 - e6', # 2 sides in foreground 'e9 + e3 - e11 - e7', 'e11 + e4 - e12 - e8', # Some elaborate loops. 'e8 - e6 + e10 + e2 - e4 - e3 - e9 + e7', 'e10 + e2 - e12 - e8 + e11 - e3 - e9 + e5', ) </syntaxhighlight> Result of each of the above calculations should be <math>0.</math> Because of small rounding errors, result may not be exactly <math>0.</math> See values_of_i and values_of_e above. This code verifies that result is close enough to <math>0</math> to be acceptable. For example: <math>\text{result} = e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5.</math> <math>\text{v1} = e_{10} + e_2 + e_{11} + e_5,</math> sum of positive values. <math>\text{status = (abs(result)} < \text{(v1}*\text{Tolerance)). status}</math> should be <math>\text{True.}</math> <syntaxhighlight lang=python> # python code import re for str1 in t1 : str2 = re.sub ('\+', ';+', str1) str2 = re.sub ('\-', ';-', str2) L1 = str2.split(';') positives = ''.join([ v for v in L1 if '-' not in v ]) result = eval(str1) v1 = eval( positives ) status = (abs(result) < (v1*Tolerance)) str3 = (str1+(' '*40))[:42] str4 = ((' '*10) + str(result))[-10:] str5 = '{}{} {}'.format(str3,str4,status) print ( str5 ) </syntaxhighlight> {| class="wikitable" |- ! Calculation || Result !! Result Acceptable |- | <math> i_1 + i_{10} - i_2 </math> | <math> -1.2E-28 </math> | <math>\text{True}</math> |- | <math> i_1 + i_3 - i_9 </math> | <math> -6E-29 </math> | <math>\text{True}</math> |- | <math> i_3 + i_{11} - i_4 </math> | <math> -3E-29 </math> | <math>\text{True}</math> |- | <math> i_{10} + i_6 - i_5 </math> | <math> 0E-29 </math> | <math>\text{True}</math> |- | <math> i_{11} + i_8 - i_7 </math> | <math> 0E-29 </math> | <math>\text{True}</math> |- | <math> i_6 + i_8 - i_{12} </math> | <math> -2E-29 </math> | <math>\text{True}</math> |- | <math> i_5 + i_7 + i_9 - i_2 - i_4 - i_{12} </math> | <math> -3E-29 </math> | <math>\text{True}</math> |- | <math> e_1 + e_2 - e_4 - e_3 </math> | <math> 0E-28 </math> | <math>\text{True}</math> |- | <math> e_5 + e_6 - e_8 - e_7 </math> | <math> 0E-27 </math> | <math>\text{True}</math> |- | <math> e_9 + e_1 - e_{10} - e_5 </math> | <math> 0E-27 </math> | <math>\text{True}</math> |- | <math> e_{10} + e_2 - e_{12} - e_6 </math> | <math> -1.3E-26 </math> | <math>\text{True}</math> |- | <math> e_9 + e_3 - e_{11} - e_7 </math> | <math> -1.3E-26 </math> | <math>\text{True}</math> |- | <math> e_{11} + e_4 - e_{12} - e_8 </math> | <math> 0E-27 </math> | <math>\text{True}</math> |- | <math> e_8 - e_6 + e_{10} + e_2 - e_4 - e_3 - e_9 + e_7 </math> | <math> 0E-27 </math> | <math>\text{True}</math> |- | <math> e_{10} + e_2 - e_{12} - e_8 + e_{11} - e_3 - e_9 + e_5 </math> | <math> 0E-27 </math> | <math>\text{True}</math> |} ===Big Matrix 64 by 65=== {{RoundBoxTop|theme=2}} [[File:0706matrix64_01.png|thumb|400px|'''Diagram of resistive network containing 64 resistors.''' </br> Positive current flow is assumed to be from bottom to top or from left to right. ]] A network containing 64 resistors is connected as shown in diagram. What is resistance R between points P1, P2? <syntaxhighlight lang=python> # python code values_of_R = ( R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, R30, R31, R32, R33, R34, R35, R36, R37, R38, R39, R40, R41, R42, R43, R44, R45, R46, R47, R48, R49, R50, R51, R52, R53, R54, R55, R56, R57, R58, R59, R60, R61, R62, R63, R64, R65, R66, R67, R68, R69, R70, R71, R72, R73, ) = [ v for v in range (100,731,10) ] </syntaxhighlight> While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64 resistors and 64 simultaneous equations to be solved. {{RoundBoxBottom}} ====Create matrix==== An examination of the circuit produces the following conditions: <syntaxhighlight lang=python> # python code iconditions = [ "+ i10 + i70 - i44" , "+ i11 - i10 - i49" , "+ i12 - i11 - i54", "+ i13 - i12 - i59" , "+ i14 - i13 - i64" , "+ i15 + i44 - i43", "+ i16 + i49 - i15 - i48 - i70" , "+ i17 + i54 - i16 - i53" , "+ i18 + i59 - i17 - i58", "+ i19 + i64 + i71 - i18 - i63" , "+ i69 - i19 - i68" , "+ i20 + i43 - i42", "+ i21 + i48 - i20 - i47" , "+ i22 + i53 - i21 - i52" , "+ i23 + i58 - i22 - i57", "+ i24 + i63 - i23 - i62" , "+ i68 - i24 - i67" , "+ i25 + i42 - i41", "+ i26 + i47 - i25 - i46" , "+ i27 + i52 - i26 - i51" , "+ i28 + i57 - i27 - i56", "+ i29 + i62 - i28 - i61" , "+ i67 - i29 - i66" , "+ i30 + i41 - i40", "+ i31 + i46 - i30 - i45 - i72" , "+ i32 + i51 - i31 - i50" , "+ i33 + i56 - i32 - i55", "+ i34 + i61 + i73 - i33 - i60" , "+ i66 - i34 - i65" , "+ i36 + i45 - i35", "+ i37 + i50 - i36" , "+ i38 + i55 - i37" , "+ i39 + i60 - i38", "+ i65 - i39 - i73" ] sx = 'len(iconditions)' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> len(iconditions) = 34 </syntaxhighlight> <syntaxhighlight lang=python> econditions = [ "+ e40 + e30 - e35 - e45" , "+ e41 + e25 - e30 - e46" , "+ e42 + e20 - e25 - e47", "+ e43 + e15 - e20 - e48" , "+ e44 + e10 - e15 - e49" , "+ e45 + e31 - e36 - e50", "+ e46 + e26 - e31 - e51" , "+ e47 + e21 - e26 - e52" , "+ e48 + e16 - e21 - e53", "+ e49 + e11 - e16 - e54" , "+ e50 + e32 - e37 - e55" , "+ e51 + e27 - e32 - e56", "+ e52 + e22 - e27 - e57" , "+ e53 + e17 - e22 - e58" , "+ e54 + e12 - e17 - e59", "+ e55 + e33 - e38 - e60" , "+ e56 + e28 - e33 - e61" , "+ e57 + e23 - e28 - e62", "+ e58 + e18 - e23 - e63" , "+ e59 + e13 - e18 - e64" , "+ e60 + e34 - e39 - e65", "+ e61 + e29 - e34 - e66" , "+ e62 + e24 - e29 - e67" , "+ e63 + e19 - e24 - e68", "+ e64 + e14 - e19 - e69" , "+ e44 + e70 - e15" , "+ e64 + e14 - e71", "+ e40 + e30 - e72" , "+ e60 + e73 - e39" ] sx = 'len(econditions)' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> len(econditions) = 29 </syntaxhighlight> In iconditions and econditions there are 63 conditions that include all values of i and e. These 63 conditions are used to create matrix. The 64th condition is added manually. The matrix is: <syntaxhighlight lang=python> matrix = [ [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,], [-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,], [0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,], [0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,], [0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,], [0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,], 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[0,0,0,0,140,0,0,0,0,-190,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,-690,0,0,0,0,0,], [0,0,0,0,0,-150,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,440,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,700,0,0,0,0,], [0,0,0,0,140,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,640,0,0,0,0,0,0,-710,0,0,0,], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,400,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,0,0,], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-390,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,600,0,0,0,0,0,0,0,0,0,0,0,0,730,0,], [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,], ] sx = 'len(matrix)' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> len(matrix) = 64 </syntaxhighlight> Bottom line of matrix is equivalent to: <math>(1)i10 + (-1) = 0</math> or <math>i10 = 1.</math> ====First result==== The following conditions are included for testing: <syntaxhighlight lang=python> more_econditions = [ "+ e49 + e70 - e10", "+ e19 + e69 - e71", "+ e35 + e45 - e72", "+ e65 + e73 - e34", ] extra_test_conditions = [ "+ e40 + e41 + e42 + e43 + e44 + e10 - e35 - e45 - e46 - e47 - e48 - e49", "+ e45 + e46 + e47 + e48 + e49 + e11 - e36 - e50 - e51 - e52 - e53 - e54", "+ e50 + e51 + e52 + e53 + e54 + e12 - e37 - e55 - e56 - e57 - e58 - e59", "+ e55 + e56 + e57 + e58 + e59 + e13 - e38 - e60 - e61 - e62 - e63 - e64", "+ e60 + e61 + e62 + e63 + e64 + e14 - e39 - e65 - e66 - e67 - e68 - e69", "+ e44 + e10 + e11 + e12 + e13 + e14 - e69 - e15 - e16 - e17 - e18 - e19", "+ e43 + e15 + e16 + e17 + e18 + e19 - e68 - e20 - e21 - e22 - e23 - e24", "+ e42 + e20 + e21 + e22 + e23 + e24 - e67 - e25 - e26 - e27 - e28 - e29", "+ e41 + e25 + e26 + e27 + e28 + e29 - e66 - e30 - e31 - e32 - e33 - e34", "+ e40 + e30 + e31 + e32 + e33 + e34 - e65 - e35 - e36 - e37 - e38 - e39", "+ i35 + i40 + i72 - i14 - i69 - i71", "+ e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14 - e35 - e36 - e37 - e38 - e39 - e65 - e66 - e67 - e68 - e69", "+ e40 + e41 + e42 + e43 + e15 + e16 + e17 + e18 + e19 + e69 - e35 - e36 - e37 - e38 - e60 - e61 - e62 - e63 - e64 - e14", "+ e40 + e41 + e42 + e20 + e21 + e22 + e23 + e24 + e68 + e69 - e35 - e36 - e37 - e55 - e56 - e57 - e58 - e59 - e13 - e14", "+ e40 + e41 + e25 + e26 + e27 + e28 + e29 + e67 + e68 + e69 - e35 - e36 - e50 - e51 - e52 - e53 - e54 - e12 - e13 - e14", "+ e40 + e30 + e31 + e32 + e33 + e34 + e66 + e67 + e68 + e69 - e35 - e45 - e46 - e47 - e48 - e49 - e11 - e12 - e13 - e14" ] </syntaxhighlight> Produce results: <syntaxhighlight lang=python> if 1 : import decimal dD = decimal.Decimal dgt = decimal.getcontext() Precision = dgt.prec = 60 # Adjust as necessary. Tolerance = dD("1e-" + str(Precision-4)) # Adjust as necessary. import re # Regular expressions. def produce_result (input, flag = 0) : thisName = 'produce_result (input, flag = {}) :'.format(flag) values_of_i = solveMbyN (input) ( i10, i11, i12, i13, i14, i15, i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30, i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45, i46, i47, i48, i49, i50, i51, i52, i53, i54, i55, i56, i57, i58, i59, i60, i61, i62, i63, i64, i65, i66, i67, i68, i69, i70, i71, i72, i73, ) = values_of_i values_of_e = ( e10, e11, e12, e13, e14, e15, e16, e17, e18, e19, e20, e21, e22, e23, e24, e25, e26, e27, e28, e29, e30, e31, e32, e33, e34, e35, e36, e37, e38, e39, e40, e41, e42, e43, e44, e45, e46, e47, e48, e49, e50, e51, e52, e53, e54, e55, e56, e57, e58, e59, e60, e61, e62, e63, e64, e65, e66, e67, e68, e69, e70, e71, e72, e73, ) = [ (i*R) for (i,R) in zip (values_of_i, values_of_R) ] E1_ = 'e40 + e41 + e42 + e43 + e44 + e10 + e11 + e12 + e13 + e14' E1 = eval(E1_) I1 = eval('i40 + i72 + i35') R1 = E1/I1 if not flag : return R1 # Check all conditions. # compiled regular expression plus, compiled regular expression minus crep,crem = [ re.compile(re.escape(v)) for v in '+-' ] # # This code verifies that result of evaluation of condition # is close enough to 0 to be called 0. # for condition in (iconditions + econditions + more_econditions + extra_test_conditions): str2 = crep.sub(';+', condition) str3 = crem.sub(';-', str2) positives = [ v for v in str3.split(';') if ('+' in v) ] pos_sum = eval(''.join(positives)) sum = eval(condition) # pos_sum - neg_sum = sum neg_sum = pos_sum - sum min,max = sorted((neg_sum, pos_sum)) status = (abs(sum) <= (Tolerance*min)) if not status : print (thisName, 'condition failed:') print (' ', condition, sum) return None return R1 # Convert all values in matrix to Decimal objects matrix = [ [ dD(str(eval(str(p)))) for p in v ] for v in matrix ] enable_reduceRow = 1 R = produce_result (matrix,1) sx = 'R' ; print (sx, '=', eval(sx)) </syntaxhighlight> <syntaxhighlight> R = 723.928575988442547699732563335265663515838075102937503680598 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 60. ====Matrix corrected==== {{RoundBoxTop|theme=2}} [[File:0710matrix64_01.png|thumb|400px|'''Diagram of network containing 64 resistors''' </br> Diagram has been corrected to show positive direction of current through R70, R73. ]] Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right. Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below. Corrected diagram conforms to assumption that all current flow is from bottom towards top or from left to right. {{RoundBoxBottom}} <syntaxhighlight lang=python> number_of_columns = 65 output = [] # # Test this matrix 64 times using # i10 = 1 # i11 = 1.1 # i12 = 1.2 # ....... # ....... # i70 = -7 # i71 = 7.1 # i72 = 7.2 # i73 = -7.3 # for I in range(10, 74): new_line = [ dD(0) ] * number_of_columns place_in_line = I - 10 new_line[place_in_line] = dD(1) i = dD(I)/10 if I in (70,73) : new_line[-1] = i else : new_line[-1] = -i matrix[-1] = new_line r = produce_result (matrix,1) if r : output += [ r ] sx = 'len(output)' ; print (sx, '=', eval(sx)) # Produce results with slightly reduced precision: dgt.prec -= 4 set2 = { r+0 for r in output } dgt.prec += 4 len2 = len(set2) if len2 == 1 : r, = set2 print ('R =', r) else : print (len2, 'values of R:') for v in set2 : print (' ', v) </syntaxhighlight> <syntaxhighlight> len(output) = 64 R = 723.92857598844254769973256333526566351583807510293750368 </syntaxhighlight> <math>\text{R}</math> is printed with precision of 56. With precision of 56 all 64 results are equal. [[Category:Equations]] azk0zp9hpuadnvx5hloasd5vm7d83br 0 276381 2834971 2573916 2026-09-29T07:07:01Z P U3270518 3106535 corrected spelling error 2834971 wikitext text/x-wiki {{title|Emotional eating:<br>What is emotional eating and what effect does it have on emotion and health?}} {{MECR3|1=https://youtu.be/R1eFbJ57oAU}} __TOC__ ==Overview== {{RoundBoxTop|theme=1}} '''Case study: the beginning''' Aurora is almost at the end of her first semester of university. She struggled to adapt to university and was feeling a lot of pressure since she was the first in her family to attend tertiary education. As she was getting closer and closer to the end, she noticed that she had put on some significant weight as compared to the start of the year. She decided to look into some options for help. {{RoundBoxBottom}} Emotional eating is the act of engaging with food when a negative emotion makes itself present (Adriaanse et al., 2011). This chapter discusses eating, emotional eating and its connection to emotions and health. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is emotional eating? * What effect does emotional eating have on emotion? * What effect does emotional eating have on health? {{RoundBoxBottom}} == Why do we eat? == [[File:Good Food Display - NCI Visuals Online.jpg|thumb|257x257px|''Figure 1''. Different types of food]] Eating is a necessary element to human survival and it has the ability to give pleasure (Gahagan, 2012). When eating is perturbed, it can lead to undernutrition, over nutrition and a wide variety of eating disorders. There is a sophisticated biological system in place to promote energy balance in the body. When there is an energy deficit, ghrelin, a peptide, is released from the stomach to help stimulate appetite. Once eating has been completed, pancreatic insulin, intestinal peptide YY and leptin are released to decrease the stimulation of appetite (Gahagan, 2012). Although this in place system exists, humans have the capacity to override this, meaning that consumption behaviours are often independent from the physiological needs (Gahagan, 2012). Factors such as altering emotional states or social settings can be a motivator to either eat or not eat (Renner et al., 2012). == What is emotional eating? == Emotional eating is often characterised as the tendency to overeat in response to negative emotions such as anxiety or irritability (Adriaanse et al., 2011). Negative emotions can induce eating because of its capability to reduce the intensity of the negative emotions (Macht & Simons, 2010). Generally, individuals are assessed for emotional eating by self-reported tests (Adriaanse et al., 2011). An example of these tests is the ''Emotional Eating Scale'' by Arnow, Kenardy & Agras or the ''Emotional Overeating Questionnaire'' by Masheb & Grilo (Adriaanse et al., 2011). A problem with these tests is that the characteristics specified in these tests differ from test to test, but a main characteristic that is recognised by many psychologists is that these individuals report that they eat not only to satisfy hunger, but additionally when they’re feeling emotional (Adriaanse et al., 2011). On one hand, psychologists acknowledge that emotional eating is pretty common with the exception that the amount of food an individual eats differs from each individual (Macht & Simons, 2010). On the other hand, some psychologists believe that emotional eating is an inappropriate reaction to negative emotions since it is also common for negative emotions to cause a loss of appetite and a decrease in food intake (Spoor et al., 2007). There are a few mechanisms and theories that are believed to instigate the cause of emotional eating. First we consider a combination of psychological mechanisms and physiological mechanisms (Macht & Simons, 2010). The psychological aspect is that during the ingestion of food, there is a response of a strong sense of pleasure that is released which improves the emotional state (Macht & Simons, 2010). The physiological aspect of this combination is that after the ingestion of food, nutrients in the food may affect neurochemical and endocrine systems that have a connection to emotions (Macht & Simons, 2010). Another mechanism would be the suggestion of poor {{what}} regulation practices (Spoor et al., 2007). Unfortunately, there are still some gaps in research in regards to the cause of emotional eating processes as the characteristics differ from case to case (Adriaanse et al., 2011). {| class="wikitable" |+Emotional Hunger vs. Physical Hunger (Smith et al., 2020) !Emotional hunger !Physical hunger |- |Comes on suddenly |Comes on gradually |- |Needs to be satisfied straight away |Can wait |- |Craves comfort foods |Open to options |- |Isnt satisfied when full |Will stop when stomach is full |- |Triggers negative feelings |Associated with feelings of satisfaction |} {{RoundBoxTop|theme=1}} '''Case study: the middle''' Upon talking to a doctor about her mysterious weight gain, Aurora was asked to take a self-reported test. As she was progressing through the questionnaire, it became clear to the doctor that Aurora had been using food as a coping mechanism to the negative emotions she was feeling throughout the semester, creating food habits she did not have previously. {{RoundBoxBottom}} == Eating and emotions == [[File:Emotional Eating Cycle.jpg|thumb|426x426px|''Figure 2''. Emotional eating cycle|left]] Emotional eating generally occurs when negative emotions arise (Adriaanse et al., 2011). There are a few theories surrounding the mechanism of poor regulation practices (Spoor et al., 2007). Psychosomatic theory states that individuals partake in emotional eating when they feel a negative emotion because they have learned that food assists in alleviating the unwanted emotion (Spoor et al., 2007). Similar to this, escape theory states that individuals partake in emotional eating because they are attempting to shift their attention from their emotional state and on to the food they are consuming (Spoor et al., 2007). A common negative emotion is stress. Stress is defined as a feeling of emotional or physical tension (A.D.A.M., 2021). It usually manifests during an event or thought that causes an individual to feel frustrated, angry or nervous (A.D.A.M., 2021). In small doses, stress can be beneficial as it is the body’s response to challenges and demands but when stress lasts a long time, it can be detrimental to the health of that individual (A.D.A.M., 2021). In regards to stress, coping is a process that is an attempt to manage stressful demands and the emotions that accompany them (Spoor et al., 2007). Three main techniques for coping with stress include a task-orientated approach, an emotion-oriented approach and an avoidance approach (Spoor et al., 2007). When an individual is feeling a great deal of stress, the body releases a high level of the stress hormone, cortisol, which cause the body to crave foods that give a burst of energy and pleasure (Smith et al., 2020). When eating becomes the primary source of emotional regulation, often individuals find themselves trapped in an emotional eating cycle (figure 2) (Smith et al., 2020). When the negative emotion arises, they turn to food for comfort (Smith et al., 2020). Unfortunately, emotional eating can become overeating and, in turn, the individual then feels guilty about the food they have consumed, causing another negative emotion to arise, kickstarting the cycle again (Smith et al., 2020). This engagement with food can cause real problems to an individual’s health (Smith et al., 2020). == Eating and health == Although emotional eating isn’t necessarily bad; however, when excessive, it can pose great risks to an individual’s health (Smith et al., 2020). For the human body to thrive, it is a necessity that it has access to the correct amount of food and nutrients (WHO, 2021). Emotional eating can cause an individual to create unhealthy eating habits such as overeating and the overconsumption of one particular food (Adriaanse et al., 2011). In some cases, these habits can lead to severe eating disorders (Adriaanse et al., 2011). Overall, emotional eating can be extremely detrimental to the health of the individual in both a physical and mental sense (Adriaanse et al., 2011). == Self-regulation == [[File:Red Tea.jpg|thumb|''Figure 3''. A cup of tea]] According to the work by Smith and colleagues (2020), the first step in learning to regulate emotional eating is recognising that the act occurs when negative emotions arise. Once this acknowledgment has been made, it is advised that whenever the need to grab usual comfort food arises, backtracking and investigating what the initial trigger could be and to take not of it for future reference. Forcing oneself to take measures such as diets will not allow the individual to properly gain control of their eating habits, instead eating habits can be best regulated by finding a distraction when the cravings become demanding. Such distractions could be: going for walks, drinking a cup of tea, calling a friend or engaging in an activity that brings you joy. These will ensure that the attention has been shifted away from food and will eventually become natural habits. It is okay to engage with the negative emotions, it is just a matter of taking that energy and putting it into something that will not cause any harm (Smith et al., 2020). {{RoundBoxTop|theme=1}} '''Case study: the end''' After having that conversation with her doctor, Aurora began taking note of all the times she felt triggered to eat. She found that when she was having problems engaging with her university content, she would feel the urge to eat. Now Aurora goes for a walk near her home whenever she feels this way and is beginning to see a significant difference in her food consumption. {{RoundBoxBottom}} ==Conclusion== Emotional eating is the act of engaging with food when a negative emotion makes itself present (Adriaanse et al., 2011). Although the act of emotional eating is instigated by emotion, emotional eating effects emotions, especially in reference to the emotional eating cycle shown in figure 1 (Smith et al., 2020; Spoor et al., 2007). It starts with the negative emotions but those emotions can become stronger at the hand of emotional eating (Smith et al., 2020). In turn, this cycle then plays a role in health (Smith et al., 2020). Emotional eating causes unhealthy eating habits that can pose a threat to the health of the individual (Adriaanse et al., 2011). It is important to acknowledge your emotions (Smith et al., 2020), find coping mechanisms that help you respond to the negative emotions (Spoor et al., 2007) and seek help if it becomes too much for you to deal with (Smith et al., 2020). ==See also== * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) *[[wikipedia:Emotional_eating|Emotional eating]] (Wikipedia) *[[Motivation and emotion/Book/2019/Junk food and psychological distress|Junk food and psychological distress]] (Book chapter, 2019) ==References== {{Hanging indent|1= A.D.A.M. (2021). Stress and your health. Medline Plus. 20 October 2021, from https://medlineplus.gov/ency/article/003211.htm. Adriaanse, M., de Ridder, D., & Evers, C. (2010). Emotional eating: Eating when emotional or emotional about eating?. ''Psychology and Health, 26''(1), 23-39. https://doi.org/10.1080/08870440903207627 Gahagan, S. (2012). Development of Eating Behavior. Journal Of Developmental & Behavioral Pediatrics, 33(3), 261-271. https://doi.org/10.1097/dbp.0b013e31824a7baa Kaye, W. (2021). Health Consequences. National Eating Disorders. 29 August 2021, from https://www.nationaleatingdisorders.org/health-consequences. Macht, M., & Simons, G. (2010). Emotional Eating. ''Emotion Regulation and Well-Being'', 281-295. https://doi.org/10.1007/978-1-4419-6953-8_17 Petersson, S., Gullbing, L., & Perseius, K. (2021). Just like fireworks in my brain – a Swedish interview study on experiences of emotions in female patients with eating disorders. ''Journal Of Eating Disorders, 9''(1). https://doi.org/10.1186/s40337-021-00371-2 Renner, B., Sproesser, G., Strohbach, S., & Schupp, H. (2012). Why we eat what we eat. The Eating Motivation Survey (TEMS). ''Appetite, 59''(1), 117-128. https://doi.org/10.1016/j.appet.2012.04.004 Smith, M., Robinson, L., Segal, J., & Segal, R. (2020). Emotional Eating. HelpGuide.org. Retrieved 29 August 2021, from https://www.helpguide.org/articles/diets/emotional-eating.htm#. Spoor, S., Bekker, M., Van Strien, T., & van Heck, G. (2007). Relations between negative affect, coping, and emotional eating. ''Appetite, 48''(3), 368-376. https://doi.org/10.1016/j.appet.2006.10.005 WHO. (2021). Nutrition. World Health Organization. 20 October 2021, from https://www.who.int/health-topics/nutrition. }} ==External links== *[https://www.befrienders.org/ Befrienders World Wide] - World Wide Helpline (befrienders.org) *[https://butterfly.org.au/get-support/helpline/ Butterfly Foundation helpline] - Australian Helpline (butterfly.org) *[https://youtu.be/8bE5XLGNPF0 Lose Weight AND Keep It Off: Emotional Eating] - TED Talk (youtube.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Eating]] dq0toiaiku78t97lydw47stvncn8uwm 0 277225 2834973 2345136 2026-09-29T07:21:31Z P U3270518 3106535 Corrected grammar error 2834973 wikitext text/x-wiki {{title|Brain fog:<br />What is brain fog, what causes it, and how can it be managed?}} {{MECR3|1=https://youtu.be/echnJgGCfwI}} __TOC__ == Overview == The purpose of this chapter is to help people understand how experiencing [[:w:Clouding of consciousness|brain fog]] may not impair an individual as much as early research seems to suggest. It is important to understand that, across many studies spanning peer-reviewed articles and longitudinal experiments, the term brain fog is imprecise, and to date the cause of this symptom is unknown. Across the breadth of five studies spanning close to four decades, all articles allude to physical fatigue contributing to mental lapses causing erratic and unbalanced trends in emotion and cognitive ability. They propose that brain fog is prevalent in many mental illnesses. A contemporary study even showcasing brain fog as a major symptom in long [[COVID-19|COVID]] (Callan, 2021). The concept of brain fog is measured to correlate the effect it has on emotion. An overload of stress may bring about brain fog and can become overwhelming for the individual to deal with simple tasks that artificially elevate in difficulty due to an inability to think clearly. Findings suggest that brain fog is a cognitive complaint like mental fatigue (Zung, 1972). Poor posture may even be associated with brain fog (Ocon, 2013). Yet this impairment is not well recognised or thoroughly understood in its depth and capacity. Before brain fog was considered a topic worth developing considerable research, it was mainly characterised as forgetfulness and/or cloudy thoughts. When establishing brain fog as a serious threat to cognition, and more broadly emotion, studies coupled the symptom of brain fog with relevant syndromes and mental illnesses {{missing}} concerned with information collection and interpretation. A variety of studies linked Chronic Fatigue Syndrome (or CFS) to brain fog as an overflow of information that can’t be mentally processed in ample time, making for lapses in decision-making related to turning information into action. Other studies look toward the framework of Cognitive Load Theory (CLT; Paas, 2003; Sweller, 2011). This theory refers to the used amount of working memory resources and studies have historically focused on CLT have often indicated that brain fog can arise when memory reserves are drained, resulting in an individual becoming easily overwhelmed. {{RoundBoxTop}}'''Focus questions''' * What is brain fog? * What causes brain fog? * How can brain fog be managed? {{RoundBoxBottom}} == What is brain fog? == Brain fog can be characterised as difficulty focusing, thinking and communicating, relating to impaired cognition and performance on cognitive tasks. In contrast, the least common descriptors (thoughts moving too quickly, detached, lost, sleepy) were more indicative of general fatigue, anxiety, and depression (Zung, 1972). Individuals subjectively experienced brain fog as mental fatigue, suggesting that the increased change in cerebral perfusion and activation during cognitive tasks was perceived as stressful and exaggerated exhaustion (De Luca et al, 1995). This means that by increasing activity in the brain, the processes in interpreting information become slower as more resources are needed to fuel the brain in this action. === Build on literature === In order to understand how cognitive impairments impact on brain fog in CFS in this case, as well as to guide future research, the goal of this chapter {{ic|cover goals in the Overview}} is to analyse the cognitive symptoms which lead to impairment to enhance the quality of research to develop high-end articles that mentions brain fog in focused literature in the future{{rewrite}}. === Distinctive features of brain fog === [[File:WorkingCogLoad.png|thumb|''Figure 1''. An example highlighting the equation of Cognitive Load Theory as it pertains to depleting mental resources.]] Brain fog can be informally defined as a temporary state of diminished mental capacity marked by the inability to concentrate or to think or reason clearly. '''Mental''' * Some individuals who suffer from brain fog could be contributed {{huh}} to a genetic disorder that causes defects in connective tissue (Rowe et.al, 1999) * Brain fog could be the result of overworked mental capacity with sufficient {{huh}} rest. * Typically characterised by having trouble concentrating and forgetful thinking. '''Physical''' * Brain fog is difficult to physically diagnose as individuals often show {{missing}} same signs and symptoms as fatigue. * Physical difficulties such as lapses in concentration and difficulty in recalling information for it to processed. == Key theory associated with brain fog == {{expand}} === Cognitive load theory === [[wikipedia:Cognitive load|Cognitive load theory]] (Sweller, 2011) is concerned with the design of methods that efficiently use people's limited cognitive processing capacity to apply knowledge and skills to interpret and relate to new situations{{example}}. Comprised of three schema-based instructional designs in relation to differential {{huh}} between intrinsic, extraneous and germane cognitive load{{gr}}: * ''Intrinsic cognitive load'' is the difficulty in which a single topic in associated with. * ''Extraneous cognitive load'' is created based on the level of difficulty the information is to process. * ''German cognitive load'' is processing the information to be then enacted and stored in autonomic process to the individual. CLT is based on a cognitive architecture that consists of a limited working memory with partly independent processing units for visual and auditory information, which interacts with an unlimited long-term memory (Paas, 2003). These functions of human cognition have been used to design a variety of novel efficient instructional methods{{example}}. The associated research has shown that measures of cognitive load can reveal important information for CLT that is not usually the sole focus of research{{vague}}. Particularly, the combination of performance and cognitive load measures have been identified to build a reliable picture of the mental efficiency of the individual. Cognitive load theory assumes that knowledge can be divided into biologically primary knowledge that we have evolved to acquire and biologically secondary knowledge that is important for cultural reasons. Secondary knowledge, unlike primary knowledge, is the subject of instruction. It is processed in a manner that is analogous to the way biological evolution processes information (Sweller, 2011). When dealing with secondary knowledge that cannot be easily recalled, human cognition becomes far more difficult, requiring a very large information store, the contents of which are acquired largely by obtaining information from other information stores. Novel information is generated by a random generate and test procedure with only very limited amounts of novel information able to be processed at any given time. This contrast can become difficult when brain fog appears as testing information stored may not directly be available, making for foggy patches in recall for the individual. In contrast, very large amounts of organized information stored in the information store can be processed in order to generate a higher-function task.{{example}} === Quiz === <quiz display=simple> {Intrinsic, extraneous and germane cognitive load comprise the schema-based instructional design of cognitive load: |type="()"} + True - False </quiz> === Peer-reviewed study === In linking CFS to brain fog, a study conducted by DeLuca et al. (1995) showed that CFS subjects have intact memory storage, consolidation, and retrieval and “higher order” cognitive function; however, they exhibit decreased speed and efficiency in information processing, especially with auditory material, and subjectively perceive a more generalized cognitive impairment (DeLuca et al., 1995). Over the past 20 years, research has answered many questions regarding cognitive symptoms in CFS (Ocon, 2013). The experience of “brain fog” in those with CFS appears to be the conscious perception of cognitive impairment and is related to mental fatigue. Interestingly, this perception of generalized cognitive impairment fits the concept of brain fog. Decreased sleep time and sleep quality have been shown to impair performance on various cognitive tasks (Deak, 2010). Individuals with CFS do not exhibit complete cognitive disability or dementia. However, they may experience deficits in working memory, information processing, and attention, which then may translate into a longer reaction time during tasks. It is logical that the perception of such deficits would be described as mental cloudiness and difficulty thinking. Drawing from an overview of the study, cognitive impairment in CFS is exacerbated by stressful stimuli such as a difficult mental task, exercise, and/or orthostatic stress. Physiologically, the cognitive impairments may be associated with impaired cardiovascular ability and altered activation of information recollection during mental tasks. Psychiatric diseases do not seem to be related to cognitive impairment. Importantly, the perception of the symptoms of CFS is often exaggerated and excessive compared to what is measured. Aside from the cognitive impairments, this perception may be disabling. Thus, brain fog is caused by measurable cognitive impairments but also the more subjective perception of mental fatigue and its impact on the individual with CFS (Ocon, 2013). Interactions of multiple factors clearly play a role. One approach is that altered information recollection are {{gr}} exacerbated by a stressor, such as orthostasis or a difficult mental task, that results in the decreased ability to readily process information, which is then perceived as fatiguing and experienced as brain fog. A more contemporary study examined how long-term symptoms of COVID-19 could possibly play a role in exacerbating brain fog (Callan, 2021). It was found that while the term brain fog was used sparingly among participants, the physical afflictions described matched prior representations of the impairment. === Case study === {{Robelbox|theme={{{theme|10}}}|title=Case study}} Sarah is a 40-year old, female teaching aide. She initially presented with feeling sick frequently, ongoing sore throat, reports a high level of tiredness all the time and persistent lapses in concentration. She is married with two children. Her mother has a history of hypertension. Routine testing revealed she had tested positive to a glandular fever antibody test. Sarah experienced conclusive symptoms relevant to brain fog. {{Robelbox/close}} == What causes brain fog? == {{expand}} === Brain fog and emotion === * Brain fog may feel like being lost in a rhetorical maze of sorts. * Brain fog associated with emotional feelings of anger, sadness and loneliness{{gr}}. * Being unable to think clearly may make an individual feel helpless, irritable and downcast. {{fact}} Table 1. Causes of Brain Fog {| class="wikitable" |+Causes of Brain Fog !Cause !Description |- |Stress |Chronic stress can increase blood pressure, weaken the immune system and trigger depressive symptoms. When the brain is exhausted due to stress, it becomes more difficult to complete simple cognitive tasks. |- |Lack of Sleep |Poor sleep quality can also interfere with how well the brain functions. Sleeping too little can lead to poor concentration and cloudy thoughts. |- |Medical Conditions |Medical conditions associated with inflammation, fatigue can also cause mental fatigue. Brain fog is a symptom of chronic fatigue syndrome (CFS), which involves persistent fatigue for longer than six months. |} Brain fog is a symptom of a number of chronic conditions. It is particularly prevalent in diseases involving inflammation and fatigue{{fact}}. Brain fog most presents itself in chronic fatigue syndrome and fibromyalgia. The most common causes of brain fog is sustained periods of chronic stress and anxiety. The [https://en.wikipedia.org/wiki/Fight-or-flight_response fight-or-flight response] gets stuck in hyper-drive and continues to exert more resources than the brain's capacity can handle. The hormone that springs the body into action, [https://en.wikipedia.org/wiki/Cortisol cortisol], keeps being produced at a higher-than-expected level. This contributes to prolonged cloudy thinking. During this part of the response, the area of the brain that stores memories is secondary function when the threat of danger is prioritised. == How can brain fog be managed? == Treatment depends on the cause. Some common interventions to improve symptoms in the short-term may involve [[:w:Intravenous therapy|intravenous therapy]], using a saline drip, using stimulant medications{{gr}}{{fact}}. An individual’s doctor may also ask that person make some healthier lifestyle changes. Such as: *Spend less time on computer and mobile phone – remind oneself to take break. * Positive thinking and affirmations. * Get enough sleep – 7-8 hours is key. * Avoid alcohol, smoking, and drinking coffee. To date, treatment of CFS has been focused on improving the physical fatigued state rather than the cognitive impairment. Meta-analyses demonstrate that cognitive behavioural therapy and graded exercise therapy effectively treat CFS in many individuals (Edmonds et al., 2004; Price et al., 2008). It would not be surprising if mental fatigue also is improved with such therapies, but future studies are necessary to formally determine this. In an experiment examining the effects of high-fat diets (HFD) on obese mice, (Lama, 2021) assessed whether the fatty acid amide Palmitoylethanolamide (PEA) could improve depressive-like behaviour. It was found that PEA, in fact, provides a benefit to drugs used in brain fog treatment therapy to limit depressive behaviours and memory deficit{{explain}}. == Conclusion == Over a range of theories and studies discussed to find out what brain fog is, what causes it and how it can be efficiently managed it is clear there is more need for research on the subject{{vague}}. In an increasingly uncertain world due to the COVID-19, brain fog appears to be a more salient subject among researchers of clinical psychology more broadly, but also on the effects it has on motivation and emotion. Over the last 20 years, there have been advancements when linking brain fog to other cognitive difficulties and even starting to link them to a major societal roadblock in the form of a global pandemic. There is still much to be discussed including the need for how it relates to motivation as there is extensive reasoning of how it confounds emotion but not motivation{{explain}}. The positive is that the concept of brain fog is still a relatively novel topic of discussion and research where the contributions made to psychology can benefit for many generations in finding out what is brain fog, what the causes are and how we manage it on a day-to-day basis{{vague}}. == See also == * [[:w:Clouding of consciousness|Brain fog]] (Wikipedia) == References == {{Hanging indent|1= Callan, C., Ladds, E., Husain, L., Pattinson, K., & Greenhalgh, T. (2021). “I can’t cope with multiple inputs”: Qualitative study of the lived experience of ‘brain fog’ after Covid-19. medRxiv. Retrieved Oct 15 2021 from: https://www.medrxiv.org/content/10.1101/2021.08.07.21261740v1 Deak, M. C., & Stickgold, R. (2010). Sleep and cognition. ''Wiley Interdisciplinary Reviews: Cognitive Science, 1''(4), 491-500. Retrieved Oct 17 2021 from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5831725/ De Lange, F. P., Kalkman, J. S., Bleijenberg, G., Hagoort, P., vd Werf, S. P., Van der Meer, J. W., & Toni, I. (2004). Neural correlates of the chronic fatigue syndrome—an fMRI study. Brain, 127(9), 1948–1957. Retrieved 17 Oct 2021 from: https://pubmed.ncbi.nlm.nih.gov/15240435/ DeLuca, J., Johnson, S. K., Beldowicz, D., & Natelson, B. H. (1995). Neuropsychological impairments in chronic fatigue syndrome, multiple sclerosis, and depression. ''Journal of Neurology, Neurosurgery & Psychiatry, 58''(1), 38-43. Retrieved 13 Oct 2021 from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1073266/ Edmonds, M., McGuire, H., & Price, J. (2004). Exercise therapy for chronic fatigue syndrome. ''The Cochrane database of systematic reviews.'' Retrieved Oct 17 2021 from: https://pubmed.ncbi.nlm.nih.gov/15266475/ Lama, A., Pirozzi, C., Annunziata, C., Morgese, M. G., Senzacqua, M., Severi, I., ... & Mattace Raso, G. (2021). Palmitoylethanolamide counteracts brain fog improving depressive‐like behaviour in obese mice: Possible role of synaptic plasticity and neurogenesis. ''British journal of pharmacology, 178''(4), 845-859. Retrieved Sep 29 2021 from: https://pubmed.ncbi.nlm.nih.gov/32346865/ Ocon, A. J. (2013). Caught in the thickness of brain fog: exploring the cognitive symptoms of Chronic Fatigue Syndrome. ''Frontiers in physiology, 4'', 63. Retrieved 4 Sep 2021 from: https://www.frontiersin.org/articles/10.3389/fphys.2013.00063/full Paas, F., Tuovinen, J. E., Tabbers, H., & Van Gerven, P. W. (2003). Cognitive load measurement as a means to advance cognitive load theory. ''Educational psychologist, 38''(1), 63-71. Retrieved 17 Oct 2021 from: https://www.tandfonline.com/doi/abs/10.1207/S15326985EP3801_8 Price, J. R., Mitchell, E., Tidy, E., & Hunot, V. (2008). Cognitive behaviour therapy for chronic fatigue syndrome in adults. ''Cochrane Database of Systematic Reviews'', (3). Retrieved 10 Oct 2021 from: https://www.cochranelibrary.com/cdsr/doi/10.1002/14651858.CD001027.pub2/full Rowe, P. C., Barron, D. F., Calkins, H., Maumenee, I. H., Tong, P. Y., & Geraghty, M. T. (1999). Orthostatic intolerance and chronic fatigue syndrome associated with Ehlers-Danlos syndrome. ''The Journal of pediatrics, 135''(4), 494-499. Retrieved 15 Oct 2021 from: https://pubmed.ncbi.nlm.nih.gov/10518084/ Sweller, J. (2011). Cognitive load theory. In Psychology of learning and motivation, 55, 37-76. ''Academic Press.'' Retrieved 15 Oct 2021 from: https://psycnet.apa.org/record/2011-17503-002 Theoharides, T. C., Weinkauf, C., & Conti, P. (2004). Brain cytokines and neuropsychiatric disorders. ''Journal of clinical psychopharmacology, 24''(6), 577-581. Retrieved October 4 2021 from: https://www.frontiersin.org/articles/10.3389/fnins.2015.00225/full Zung, W. W. (1972). The depression status inventory: an adjunct to the self-rating depression scale. ''Journal of Clinical Psychology.'' Retrieved October 1 2021 from: https://psycnet.apa.org/record/1974-11342-001 }} == External links == {{ic|Use alphabetical order}} * [https://www.youtube.com/watch?v=zzjNlL2SdUU&ab_channel=MayoClinic The Mayo clinic q&a podcast: brain fog (YouTube)] *[https://www.youtube.com/watch?v=XOU2ubWkoPw&t=58s/ How to fix the exhausted brain] (Ted Talk) * [https://www.psychologytoday.com/au/blog/happiness-is-state-mind/202107/memory-and-brain-fog-part-1 Memory and Brain Fog] (PsychologyToday) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Brain]] [[Category:Motivation and emotion/Book/Confusion]] c3rwir9htzn5dxhgdck5j80ev8fsmn7 2 289273 2834877 2834801 2026-09-28T19:21:55Z Dc.samizdat 2856930 2834877 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space at velocity <math>c</math> on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 99c5p7v2juzdxf1x7vs7xefkutycz82 2834879 2834877 2026-09-28T19:28:44Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834879 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space at velocity <math>c</math> on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 2fhiukfl5zrdrgofmkdaioailc6tgk6 2834881 2834879 2026-09-28T19:42:08Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834881 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on helical trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are helical screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space at velocity <math>c</math> on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 4efpucb9ouayv5ij29897rkg551aqww 2834882 2834881 2026-09-28T19:43:34Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834882 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space at velocity <math>c</math> on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 2fhiukfl5zrdrgofmkdaioailc6tgk6 2834884 2834882 2026-09-28T19:46:22Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834884 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space at velocity <math>c</math> on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 1fic1qk17naf0exljxyqtdcl5yrmvqg 2834890 2834884 2026-09-28T19:58:40Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834890 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space at velocity <math>c</math> on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} o6eohy4nc15gwri1lt8z0g6uaj6uy7p 2834893 2834890 2026-09-28T20:07:45Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834893 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9s7if57w3jgeq84evn0ix6k87w5mazy 2834894 2834893 2026-09-28T20:08:52Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834894 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9uml9hz2gckyw1odwivueszqtn3mmrt 2834899 2834894 2026-09-28T20:15:40Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2834899 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional "space" because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} d1a2qddboa3y9av8w7dzno4002ye5la 2834901 2834899 2026-09-28T20:30:32Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834901 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction. This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} sok6iou60j0zjqvjuk7zuqdqwhc2i3e 2834903 2834901 2026-09-28T20:41:20Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834903 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity <math>c</math> through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} oru8jf7cgpcwrs1itjgn3r6stn6aejh 2834904 2834903 2026-09-28T20:49:16Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834904 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity <math>c</math>.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 74x8m8si6khmjn7wkrdey6l85wvrg63 2834906 2834904 2026-09-28T20:50:38Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834906 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity <math>c</math> relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to <math>c</math>, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity <math>c</math> through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} kywbwitnwu44ny8hzkknmsfcpifsmfo 2834907 2834906 2026-09-28T21:07:18Z Dc.samizdat 2856930 2834907 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is <math>2c</math> when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} excfn2ycs6u2zonfxdemwdxaobxkojo 2834908 2834907 2026-09-28T21:08:56Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834908 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} p8ixdaxvnisgwhrrav1c6t0mfm2bvtq 2834910 2834908 2026-09-28T21:17:54Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834910 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time<br> <br> ...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} d9nrsag1hllnn59vq8rfa99go6uve4o 2834912 2834910 2026-09-28T21:20:55Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834912 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions. In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} g9ybafzrlq4r0h0h3dnzm2ff4zzw1rs 2834913 2834912 2026-09-28T21:24:16Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834913 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ccd65sgg4hq0ns5reionswby8199cgj 2834914 2834913 2026-09-28T21:28:31Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834914 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time<br> <br> ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9152xtdozlnfs9zbhxstk94gedmw4ll 2834916 2834914 2026-09-28T21:45:16Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834916 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. More generally, our model is based on the geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 17dbcv7o48kxo01j3btzcf98lxwbdf2 2834919 2834916 2026-09-28T22:13:42Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834919 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion. More generally, our model is motivated by deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} sizswtycxr5d4lh9wjy7snozoxzgpa2 2834920 2834919 2026-09-28T22:17:08Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834920 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is motivated by deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} gjtnexc5p3yv47716ekprqqt7g5bsip 2834921 2834920 2026-09-28T22:19:35Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2834921 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is also an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 6azozf7zar6yca5pqjvyznvovsm41na 2834923 2834921 2026-09-28T22:32:58Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2834923 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as fast as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} c0gwpwqz96nz95utbc1jcqd70a1z2u3 2834925 2834923 2026-09-28T22:36:19Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2834925 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's geometric laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} e7k8s7y4t60raq6xxshl9x77pox9ual 2834927 2834925 2026-09-28T22:43:09Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2834927 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's kinematic laws of motion in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} s3m74g81vz4u6al3qmn9256z8eux3i1 2834928 2834927 2026-09-28T22:44:05Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2834928 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the four-dimensional principle of relativity. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} qhx8orflwb2fqo10192utv731a28o99 2834934 2834928 2026-09-29T00:07:54Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2834934 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in four-dimensional Euclidean space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 3y7h9wangi0ang9w2sumxtt7m27ctg8 2834940 2834934 2026-09-29T00:58:07Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2834940 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} agkm226znzn2scablkbku17s78qts1d 2834942 2834940 2026-09-29T01:03:39Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2834942 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space of four orthogonal spatial dimensions]] <math>\mathbb{R}^4</math>. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. More generally, our model is related to deep geometric properties of number elucidated by Coxeter and other geometers.{{Efn| ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. In a universe which is four-dimensional Euclidean space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>(n+1)</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean ''n''-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only much closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has ''two'' [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in the Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity. Symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} dilhtjnwef2ogbauzddg8gkqswv2wtr 0 298898 2834969 2755077 2026-09-29T06:52:05Z P U3270518 3106535 Corrected grammar error 2834969 wikitext text/x-wiki {{title|Multiple selves, motivation, and emotion:<br>How does the concept of multiple selves impact our motivational and emotional experiences?}} {{MECR3|1=https://youtu.be/WxkZXwvfacM}} __TOC__ ==Overview== [[File:ColorfulMinds.jpg|thumb|Figure 1. We each have different dimensions to ourselves as we navigate different situations.]] Multiple selves can be linked to many different disciplines of Psychology. In Skinner's understanding that we as humans react to stimuli from the environment around us, from Freud's theory of a multi-layered self. In simplest terms, multiple selves theory looks at how one person can be a psychologist, a scholar, a mother, a friend, and/or a daughter and how each of these interact with each other. This chapter summarises psychological theory and research about this topic and discusses how multiple selves can affect motivation and emotion within an individual. To help further understanding of the individual perspective this chapter will also address a case study conducted by Džinović (2022). In which 25 total, 11 successful and 14 unsuccessful students were selected from a prior focus group to gain rich qualitative information about multiple selves theory in terms of success, polyphonic identity and begin conceptualising opposing experiences that may occur during schooling{{gr}}. Suggestion: use a scenario or question the reader to make the overview more engaging {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is Multiple Selves Theory? * What are the implications of Multiple Selves Theory? * How is Multiple Selves theory relevant to Motivation and Emotion? {{RoundBoxBottom}} == Understanding multiple selves theory == In the basics of psychology and personality studies we are taught that the self is important to stability. More specifically, seeing yourself as a whole single person with your own traits, abilities, thoughts and feelings, and self-chosen actions is essential for positive psychological well-being (Džinović, 2022). This idea brings forth that a continuous personality should exist, evolving through personal experience but not changing completely to form fragmented selves. This, however, is unrealistic when seeing the different facets we face on a day to day basis and the different environments we must adapt and react to (Lester, 2012). This is the basis of multiple selves theory. Multiple selves theory is the understanding that each person has multiple different identities formed through connections to those around them and the groups/communities they identify with. Within social psychology, these different identities would be seen as part of the self-schema {{gr}} some of which would be seen as core parts of an individual’s identities, others could be aschematic traits, identities that are linked to the person but that the individual may not personally associate themselves closely with or be unaware of an association at all (Alós-Ferrer & Strack, 2014). Understanding multiple selves theory helps us understand how we navigate different environments in our lives. For example the way you may behave in a social setting would be completely different to how you would conduct yourself in a work environment. Multiple selves theory seeks to explain this through identifying the different parts of a person’s identity and what environmental triggers cause these shifts within a person (Karpen, 2022). An alternative to understanding multiple selves theory is through the understanding of the polyphonic self, a concept that instead of seeing through the lens of a self-schema we instead see the different parts of a person as their different ‘voices’. Each of which are individual and serve different purposes and different points of view, a key part of the polyphonic self is that each voice represents a different aspect of an identity and are aware of the other voices. The awareness of other voices is a core part of the polyphonic approach to multiple selves theory because it helps explain how a personality can be different when environmental triggers shift it without disrupting the psychological well-being of a person. Due to the awareness of different voices the different aspects of a person can interact with each other and come to joint conclusions. For example, even within a completely social setting, voices associated with a ‘work personality’ or a ‘school personality’ still hold influence and speak with the social voice to help come to conclusions.{{fact}} === Application of theory through Džinović (2022) === Both self-schemas and the polyphonic self{{gr}} are applicable when looking at Džinović’s (2022) study. Students from a vocational secondary school in Belgrade were interviewed on why they made specific choices when studying and their answers were then compared to their achieved grades. In this study ‘the real me’ is defined as the dominant voice in a person's polyphonic self and other voices were analysed. Voices from influential people in participants' lives, such as parents, held weight on the importance students saw in their success and held positive influence on their grades. More clearly, participants when interviewed would state that they would recall their parent’s ideals while studying, telling them that they needed good grades in order to have a good life which motivated their ‘real voice’ to do better. Additionally, concepts such as ‘the lazy me’ were looked at. Students would sometimes report a voice telling them not to study so much. Although this would disagree with other voices including that of ‘the real me’ it had the chance of undermining the position of power ‘the real me’ holds as the dominant voice. An explanation on why this may occur is that ‘lazy’ is within the self-schema of a person, even if it is an aschematic trait. === Quiz === <quiz display=simple> {An aschematic trait is closely related to a person's core identity. |type="()"} - True + False </quiz> == A brief overview of motivation and emotion == In order for us to talk about the effects of multiple selves on motivation and emotion we must first understand these two concepts from a psychological perspective. Motivation from a psychological perspective helps us understand why behaviour occurs, why it persists and what influences this. Understanding this is essential, without motivation why would we do anything? There are different types of motivation that are dependent on the outcome expected from the behaviour being motivated. Motivation will vary greatly for each person, meaning that the behaviour produced will also vary in the likelihood of it occurring, the success of the behaviour and its likelihood to occur again, all three of which also hold influence over each other (Reeve, 2018). In simplest terms, motivation is the basis to understanding all behaviour and as a result shapes all behavioural sciences, especially within the psychology distinction. Emotions from a psychological perspective are patterns of neural activity that shape the way we interact with ourselves and the world around us. They occur as a reaction to important life events and help us adapt accordingly to this through generating feelings, arousing bodily responses, creating motivational states and producing facial expressions. Emotions are more than what we first classify as feelings, this being just one facet of the multidimensional reality of emotions, due to this defining simply what emotions are is hard to simplify. Emotions can generate goals and, as a result, motivation. However, due to the complexity and diversity of emotions these goals differ greatly depending on the emotion generated by a life event due to the differences in the four activated aspects of emotions (Reeve, 2018). Furthermore, while certain life events may have an expected emotional response different people react differently in terms of emotions so behavioural responses from life events cannot always be predicted (Asano, 2019). For further information on this the [[motivation]] and [[emotion]] chapter’s from 2010 goes into depth about these two concepts within psychology. === Key points === * Motivation is the source behind all behaviour * Emotion is multifaceted and more complex than just feelings == Multiple selves and motivation == Behaviour is the foundation of all motivation; without motivation, there would be no reason to act. So how do multiple selves influence the motivation of a person? McConnell et al. (2012) explain that each individual self within the whole person will have its own needs, goals, and motives. If the environment where a self is needed does not strictly relate to achievement than that will not be a motivating factor within that self. This in itself helps us better understand why multiple selves start to form in the first place. With motivation being the beginning of all behaviour and we as humans being nothing without behaviour the different motivational needs dependant on the situation and environment make multiple selves essential.{{fact}} While there may be central goals and motivations for the self as whole the separate areas of a person's{{gr}} life and multiple selves within this will then have separate goals and motives as a result. Reiterating this point with the polyphonic self, the different voices one has will have different motives behind why they seek to behave a certain way. Behaviours that someone would not follow in academic, or work environments may seem rational in social settings due to the motivation of peer approval and continued acceptance within a group. {{RoundBoxLeft}} Additionally, consider aschematic traits. Would these influence your motivation? If we take being lazy as an aschematic trait of someone then it may not be realised as a motivation when deciding to do something but due to it being tied to the self-schema (or a voice within the polyphonic self) it would be a facet in determining if a behaviour occurs or not. {{LeftRightBoxClose}} === Džinović (2022) and motivation === When looking at the study done by Džinović (2022) motivation played a large role in the success of students. Those who saw themselves as high achieving students and had largely influential voices within their polyphonic self that encouraged them to achieve high grades would more often do so. The dominant voice being influenced highly by the motivation of achievement within these students helped them to not only study in the first place but to continue to study. This further builds up a tendency within the academic self of the participant and further solidifies the part of their self-schema related to schooling and achievement. === Key points === * Motivation can vary depending on which self is relevant in an environment * Each self has its own needs, goals, and motives * There are central goals and motivations for the dominant voice == Multiple selves and emotion == Now let’s look at multiple selves with our knowledge of emotions. Looking at multiple selves theory in terms of different facets of a person formed through self-schema raises the question: will people react differently emotionally depending on which facet they are in at the time of the life event occurring?  Alós-Ferrer and Strack (2014) discuss that the environment a person is shaping to does not affect emotional variability but rather what aspect of their life is being affected by the event creating the emotion to begin with. As people are see different parts of themselves as more intrinsically tied to their core self the emotional reactions vary drastically as a result. Moreover, this can drastically change not only their emotional response but also their goals, perceptions, and actions following the life event. For example, would you react to the same life event occurring within your workplace as you would within your close group of friends? In addition to this emotions typically tied to personality can vary greatly within a person when looking at multiple selves in a faceted point of view. Someone who may be seen as reserved and shy by their family may be seen as outgoing and boisterous by their friends. Both of which are true but dependant on environmental stimuli and the different selves of the person.{{fact}} If we apply the polyphonic selves approach to emotional responses within multiple selves you find a similar response. Emotions can vary greatly, this time determined on which voices hold the greatest influence on the dominant self. Emotional responses are dependant on the four facets, some of which will not change between situations (e.g., arousal), and depending on the situation, different voices may favour different emotional responses (Džinović, 2022). {{RoundBoxLeft}} === Application === To put this into context imagine a student who does volleyball on the weekends. This student is used to getting high marks and does volleyball as per their parents requested they do a sport but they personally are not that interested in it. Their self-schema would have being a high achieving student intrinsically linked to them but being part of the volleyball team is a less important trait, potentially even an aschematic trait. Due to this their reaction to failing a test or losing{{sp}} a volleyball game will be extremely different. {{LeftRightBoxClose}} === Key points === * Each self does not have its own set of emotions * Emotion prevalence changes depending on the self that is affected * Emotional intensity and variability changes depending on how intrinsically associated a self is to the dominant voice ==Conclusion== Understanding multiple selves theory helps us understand how we navigate different environments in our lives. This chapter focussed on the theories regarding self-schema and the polyphonic self but there are many more theories that help us understand multiple selves, why they exist and how they help us navigate our lives. Motivation and emotion vary depending on which self is present as each self has its own needs, goals and motives. Additionally, some selves are seen as more intrinsically associated with the dominant self and more important for achieving core system goals. Motivation and emotion helps us better understand the existence of multiple selves as it helps explain why we are more focussed on specific aspects of our lives and why we react differently. <quiz display=simple> {Do the multiple selves influence motivation and emotion? |type="()"} - Motivation is influenced but emotion is not. - Emotion is influenced but motivation is not - Neither are influenced and stay the same across the multiple selves + Both are influenced and differ accordingly </quiz> ==See also== * [[Motivation and emotion/Book/2021/Self-schemas and motivation|Self-schemas and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Textbook/Motivation/Self-concept|Self-concept]] (Textbook chapter) * [[Motivation]] (Book chapter, 2010) * [[Emotion]] (book chapter, 2010) ==References== {{Hanging indent|1= Alós-Ferrer, C., & Strack, F. (2014). From dual processes to multiple selves: Implications for economic behavior. ''Journal of Economic Psychology'', ''41'', 1-11. https://doi.org/10.1016/j.joep.2013.12.005 Asano, K. (2019). Emotion processing and the role of compassion in psychotherapy from the perspective of multiple selves and the compassionate self. ''Case Reports in Psychiatry'', ''2019'', 1-6, Article 7214752. https://doi.org/10.1155/2019/7214752 Džinović, V. (2022). The multiple self: Between sociality and dominance. ''Journal of Constructivist Psychology'', ''35''(1), 199-217. <nowiki>https://doi.org/10.1080/10720537.2020.1805063</nowiki> Karpen, S. C. (2022). The multiple selves framework in transformative learning. ''Journal of Transformative Education'', ''20''(4), 268-280. <nowiki>https://doi.org/10.1177/15413446211051278</nowiki> Lester, D. (2012). A multiple self theory of the mind. ''Comprehensive Psychology'', ''1'', 2-9. https://doi.org/10.2466/02.09.28.CP.1.5 McConnell, A. R. (2011). The multiple self-aspects framework: Self-concept representation and its implications. ''Personality and Social Psychology Review'', ''15''(1), 3-27. https://doi.org/10.1177/1088868310371101 McConnell, A. R., Shoda, T. M., & Skulborstad, H. M. (2012). The self as a collection of multiple self-aspects: Structure, development, operation, and implications. ''Social Cognition'', ''30''(4), 380-395. https://doi.org/10.1521/soco.2012.30.4.380 Reeve, J. (2018). ''Understanding motivation and emotion ''(7th ed.). Wiley. }} ==External links== * [https://dictionary.apa.org/multiple-selves APA Dictionary of Psychology] - American Psychological Association * [https://bigthink.com/thinking/selfless-book/ “The self” doesn’t exist. Instead, you constantly shape multiple selves] - Brian Lowery [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self]] az42pxkyz0y5qgfytmwg1tph9j9ecn8 0 301040 2834979 2834555 2026-09-29T07:31:48Z D.H 52339 /* Perspective of the traveler */ + 2834979 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life." |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> all the time in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] qm9clektndv23ye6xwf1q0i7jf413dw 2834981 2834979 2026-09-29T07:33:55Z D.H 52339 /* Historical references */ + 2834981 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life." |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> all the time in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] 8pg26tdth1hb9x7mn82lm1cvtr350u5 2834982 2834981 2026-09-29T07:34:22Z D.H 52339 /* Maximal proper time */ + 2834982 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 /><ref name=becqu2 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life." |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> all the time in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] e784rzc12ulj0wnis5wz6nlgygmgj8x 2834985 2834982 2026-09-29T07:37:25Z D.H 52339 /* Perspective of the traveler */ 2834985 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 /><ref name=becqu2 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that "nobody who is seriously interested in relativity should neglect to read that book".<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life. |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> all the time in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] g0ntyuxhnc95md4g02doegeuzxrxk32 2834988 2834985 2026-09-29T07:39:43Z D.H 52339 /* Perspective of the traveler */ 2834988 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 /><ref name=becqu2 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that "nobody who is seriously interested in relativity should neglect to read that book".<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life. |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> all the time in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in the ''Earth frame'' S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in the ''rocket frame'' S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] m1prvxzaq4zl7pjvil10dsjuzt0tzaj 2834990 2834988 2026-09-29T07:42:20Z D.H 52339 /* Perspective of the traveler */ 2834990 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 /><ref name=becqu2 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that "nobody who is seriously interested in relativity should neglect to read that book".<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' (with <math>\Phi</math> as potential difference) in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life. |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> all the time in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in the ''Earth frame'' S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in the ''rocket frame'' S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] 2yksb1uaumevrul2ox8xsg9lyfp5wc8 2835002 2834990 2026-09-29T08:15:27Z D.H 52339 /* Perspective of the traveler */ 2835002 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 /><ref name=becqu2 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that "nobody who is seriously interested in relativity should neglect to read that book".<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' (with <math>\Phi</math> as potential difference) in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life. |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in the ''Earth frame'' S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in the ''rocket frame'' S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates, when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] adxn26hvb3lnpe8pvu3qozse1ww7eld 2835009 2835002 2026-09-29T08:41:30Z D.H 52339 2835009 wikitext text/x-wiki {| style="width:20%; font-size:13px;" align=right |{{../Other Topics (header)}} |} ==Early history of the twin paradox== {{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 26 September 2026}}}} a) When was the [[:w:twin paradox]] applied to life forms and human beings? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /><ref group=S name=during /> report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. :*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well. b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> mention Langevin (1911), Laue (1911). :*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13). c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox? :*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922). d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Einstein (1911), Laue (1913). :*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913). e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox? :*Historical accounts<ref group=S name=debs /><ref group=S name=alizzi /> date it back to Lange (1927) and Lord Halsbury (1957). :*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22). f) Who was the first to use acceleration as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref name=cuvaj group=S /><ref group=S name=pes /> mention Langevin (1911), Einstein (1918). :*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20). g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator? :*Historical accounts<ref group=S name=miller /><ref group=S name=pes /> mention Laue (1911-13). :*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918). h) Who was the first to describe the perspective of the traveler? :*Historical accounts<ref group=S name=miller /><ref group=S name=beng /> mention Langevin (1911), Lorentz (1914), Einstein (1918). :*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921). i) Who was the first to describe a round-trip experiment in curved spacetime? :*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922). j) Who was the first to interprete the twin paradox in terms of an aether or absolute space? :*See section {{slink||Aether interpretations}} with the contribution of Langevin (1911), Wiechert (1911), Lechalas (1912). k) Who was the first to denote the round-trip experiment as paradoxical? :*Historical accounts<ref group=S name=miller /><ref group=S name=during /> point to Laue (1911). :*See section {{slink||Paradoxical?}} for details. l) Who was the first to misunderstand the twin paradox? :*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912). m) What were Einstein's contributions? :*See section {{slink||Einstein's contributions}}. ==Human beings in 1911== {| class="wikitable" style="background-color:white;{{text default color}};" ![[w:Albert Einstein|Einstein]] |- |{{anchor|Einstein 1905}}In 1905<ref name=einstein05 /> he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately <math>\tfrac{1}{2}t(v/V)^{2}</math> at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). {{anchor|Einstein 1911-HU}}In a lecture given on January 1911<ref name=einstein11a /> (published in November), he extended this "funny" (German: drollig) experiment to living organisms: {| ! width=55% | Einstein wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! | style="padding: 0px 20px 0px 20px;" |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! |} {{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}} |- !{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]] |- |He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]" published on 28 April 1911,<ref name=lammel /> including additional details. Regarding the round-trip clock experiment he wrote: {| ! width=50% | Lämmel wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. | style="padding: 0px 20px 0px 20px;" |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies. |} Lämmel in December 1920 (published 1921)<ref name=lammel2 /> again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive. {{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}} |- !{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]] |- |On 10 April 1911, published July 1911,<ref name=langevin1 /> he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: {| ! width=50% | Langevin wrote ! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]] |- | style="padding: 0px 20px 0px 20px;" |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. | style="padding: 0px 20px 0px 20px;" |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. |} {{Lorentzbox|Text=Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}} |- !{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]] |- |In lectures on 25 March and 23 May 1911, submitted July and published September 1911,<ref name=wiechert11 /> he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers: {| ! width=50% | Wiechert wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd <math>\sqrt{1-v^{2}/c^{2}}</math> ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. | style="padding: 0px 20px 0px 20px;" |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have <math>\sqrt{1-v^{2}/c^{2}}</math> throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years. |} {{Lorentzbox|Text=a) Wiechert (1915)<ref name=wiechert15 /> later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks ("Einstein's clock experiment"), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used "humorist" examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature. b) The dates given by Wiechert (1915) are not complete. The correct ones are: *Langevin's lecture on 10 April 1911, published in July. *Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September. *He was still unaware of Einstein's lecture from January 1911, published in November 1911.}} |- !{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]] |- |The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper "[[w:Berliner Tageblatt|Berliner Tageblatt]]" on 16th and 23rd October 1911,<ref name=muller /> in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote: {| ! width=50% | Müller wrote ! English translation |- | style="padding: 0px 20px 0px 20px;" |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. | style="padding: 0px 20px 0px 20px;" |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned. |} {{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}} |} ==Twins from 1911 to 1920== We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began: {| class="wikitable" style="background-color:white;" |- ! Author !! Date !! Description |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> |1911 |Two life forms that begin their life at the ''same time'' (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one. |- |[[w:Paul Gruner|Gruner]]<ref name=gruner /> |1912 |Two persons of ''same age'' (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one. |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> |1913 |The moving life form returns younger than its ''former agemates'' (German: "ehemaligen Altersgenossen"). |- |[[w:Hermann Weyl|Weyl]]<ref name=weyl /> |Easter 1918 | {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. |Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. |} {{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}} |- |[[w:Albert Einstein|Einstein]]<ref name=einstein20 /> |1920/21 |{{Anchor|Einstein 1921-TW}} {| ! width=50% | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. |If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. |} {{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}} |} ==Maximal proper time== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]] 1911 |{{anchor|Langevin 1911-PT}}In April 1911 (published July),<ref name=langevin1 /> he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind. In October 1911 (published 1912),<ref name=langevin2 /> Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' <math>R</math> than the one that stayed in an inertial frame, which is defined by the equation: :<math>\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\ \left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right] \end{matrix}</math> |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> Lectures March-May 1911 submitted July published September |{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals <math>\Delta\tau </math> of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: :<math>\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}</math> |- |[[w:Eduard Study|Study]]<ref name=study /> June 1911 |Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time. {{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}} |- |[[w:Max von Laue|Laue]] 1911-13 |{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),<ref name=laue1 /> Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.'' {{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,<ref name=laue1 /> Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: :<math>\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\tfrac{1}{c}\int_{1}^{2}du</math> {{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),<ref name=sommerfeld /> [[w:Hermann Weyl|Weyl]] (1918),<ref name=weyl /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Jean Becquerel|Becquerel]] (1922).<ref name=becqu1 /><ref name=becqu2 />}} |} ==Triangle inequality== {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Alfred Robb|Robb]] 1914-1920 |{{anchor|Robb 1914-TR}}In 1914<ref name=robb1 /> he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by <math>{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}</math>). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. {{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}} [[File:RobbTriangle.svg|right|150px]] In 1920<ref name=robb2 /> Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates :<math>\begin{matrix} & x & y & z & t\\ A\ & 0 & 0 & 0 & 0\\ B\ & 0 & 0 & 0 & 10\\ C\ & 4 & 0 & 0 & 5 \end{matrix}</math> which he plugged into :<math>\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}</math> from which he obtained the sides AB=10, AC=3, CB=3 and the inequality <math>AC+CB<AB</math>. |- |[[w:Arthur Eddington|Eddington]]<ref name=edding2 /> 1922 |He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. |- |Rogers<ref name=rogers /> 1922 |He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation <math>\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}</math>, where <math>\cosh C</math> denotes the unit-scalar product of the vectors CA, CB, and <math>\alpha,\beta,\gamma </math> the real and positive intervals BC, CA, AB. Since <math>\alpha>\beta </math> and <math>\cosh C>1</math>, it follows that <math>\alpha>\beta+\gamma </math>. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum". |} ==Negligibility of proper acceleration== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |style="vertical-align:top" |[[w:Albert Einstein|Einstein]] 1905-1918 |In 1905,<ref name=einstein05 /> Einstein used velocity time dilation <math>\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}</math> to derive the retardation of a clock performing a round-trip with constant speed <math>v</math> along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. {{anchor|Einstein 1911-VA}} In 1911 (published 1912),<ref name=einstein3 /> Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path. {{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,<ref name=einst12manu /> he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. {{anchor|Einstein 1914a-VA}}In a letter from April 1914,<ref name=einstpetz /> Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path. {{anchor|Einstein 1914b-VA}}During a conversation in May 1914,<ref name=rowe group=S /> Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with {{slink||Einstein 1914b-AC}}) {{anchor|Einstein 1918-VA}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918,<ref name=einstein18 /> Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}}) |- |[[w:Emil Wiechert|Wiechert]]<ref name=wiechert11 /> 1911 |{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities <math>+u</math> and <math>-u</math> for a long time, while A is brought after a short time from relative velocity <math>+u</math> to relative rest at which it remains a long time, and then it is brought to relative velocity <math>-u</math> for a short time. {{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}} |- |[[w:Max von Laue|Laue]]<ref name=laue3 /> 1913 |{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion. {{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)<ref name=thirring /> or [[w:Max Born|Born]] (1921).<ref name=born />}} |- |[[w:Hendrik Lorentz|Lorentz]]<ref name=lorentz1 /> 1913 |He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. {{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.<ref name=pauli />}} |} ==Relay (three brothers) experiment== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]<ref name=gbaum /> 1911 |He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. {{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}} |- |style="vertical-align:top"|[[w:Emil Wiechert|Wiechert]] 1920-1922 |In 1920 (published 1921),<ref name=wiechert20 /> Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. [[File:Wiechert1922a.png|180px|right]] In 1921 (published 1922),<ref name=wiechert21 /> Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies <math>B_{1}</math>, <math>B_{2}</math>, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. {{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers<ref name=debs group=S /> such as by Lange (1927)<ref group=S name=lange /> in which the brothers synchronize their times when they pass each other (“three brother experiment”).}} |} ==Acceleration as asymmetry indicator== While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle. {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". |- |[[w:Arnold Sommerfeld|Sommerfeld]]<ref name=sommerfeld /> 1913 |After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion". |- |[[w:Hendrik Lorentz|Lorentz]] 1913<ref name=lorentz1 /> |After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers. |- |style="vertical-align:top"|[[w:Albert Einstein|Einstein]] 1914-1920 |{{anchor|Einstein 1914b-AC}} During a conversation in 1914,<ref name=rowe group=S /> Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind ("accelerated motions are absolute"). {{anchor|Einstein 1918-AC}}In his famous "Dialog about Objections against the Theory of Relativity" from 1918<ref name=einstein18 />, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." {{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:<ref name=einstein20 /> {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. | style="padding: 0px 20px 0px 20px;" |In the case of these two twins," Einstein declared, "we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." |} In a discussion from 1922,<ref name=morand /> Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't. |} ==Frame distribution as asymmetry indicator== Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines. {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |style="vertical-align:top"|[[w:Max von Laue|Laue]] 1911-1913 |{{Anchor|Laue 1911/12-VA}} In 1911/12,<ref name=laue1 /> he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}). In 1912/13,<ref name=laue2 /> he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}). In 1913<ref name=laue3 /> Laue pointed out: {| !style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.) | style="padding: 0px 20px 0px 20px;" | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..) |} |- |[[w:Werner Bloch|Bloch]]<ref name=bloch /> September 1918 |{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames. |} ==Perspective of the traveler== {| class="wikitable" style="background-color:white;" |- ! Author !! Early examples |- |[[w:Paul Langevin|Langevin]]<ref name=langevin1 /> 1911 |{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. {{Lorentzbox|Text=Langevin used <math>v=c\left(1-\tfrac{1}{20000}\right)</math>, producing Lorentz factor <math>\gamma\approx100</math> and Doppler factor <math>\sqrt{\tfrac{c+v}{c-v}}\approx200</math>.}} |- |[[w:Hendrik Lorentz|Lorentz]] Lectures published in 1913<ref name=lorentz1 /> Similar treatment in 1914<ref name=lorentz3 /> |{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of <math>\sqrt{1-v^{2}/c^{2}}</math> with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of <math>\sqrt{\tfrac{c+v}{c-v}}</math> which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. {{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that "nobody who is seriously interested in relativity should neglect to read that book".<ref name=einstlor /> [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.<ref name=pauli />}} |- |style="vertical-align:top"| [[w:Albert Einstein|Einstein]] 1916-1920 |{{anchor|Einstein 1916-EP}}In a lecture from 1916,<ref name=einstein16 /> of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of [[w:general relativity]]." {{anchor|Einstein 1918a-EP}}In a letter from September 1918,<ref name=einadl /> Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. {{anchor|Einstein 1918-EP}}In his famous "Dialog about Objections against the Theory of Relativity" from November 1918,<ref name=einstein18 /> aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity <math>-v</math>, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity <math>+v</math>, (4) it moves with velocity <math>+v</math>, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. {{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,<ref name=einstein18b /> in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' (with <math>\Phi</math> as potential difference) in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). {{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,<ref name=einstein20 /> that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the "proper" description in terms of general relativity is as follows: {| ! style="width:50%" | German original ! English translation |- | style="padding: 0px 20px 0px 20px;" | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. | style="padding: 0px 20px 0px 20px;" | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life. |} {{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),<ref name=bloch2 /> [[w:Wolfgang Pauli|Pauli]] (1921),<ref name=pauli /> [[w:August Kopff|Kopff]] (1921),<ref name=kopff /> [[w:Karl Bollert|Bollert]] (1921),<ref name=bollert1 /> [[w:Max Born|Born]] (1921),<ref name=born /> expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of ''special'' relativity.<ref name=weiss group=S />}} |- |[[w:Hans Thirring|Thirring]]<ref name=thirring /> April 1921 |{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]] He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of <math>t'=\gamma\left(t-vx/c^{2}\right)</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation <math>t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)</math> leading to the effect of clock desynchronization, where <math>t'_{0}</math> is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. He went on to show the analogy to Einstein's explanation employing the equivalence principle, because Thirring's method gives an elementary explanation of the gravitational redshift at locations of lower potential since clocks at those locations fell behind due to desynchronization. {{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by <math>t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}</math> with <math>l=vt</math> as position of turnaround.}} |- |[[w:Paul Langevin|Langevin]]<ref name=morand /><ref name=nord /> 1922 |{{anchor|Langevin 1922-DS}}He discussed a round-trip experiment with a train traveling away and back again to the station, by demonstrating the consequences of the temporal Lorentz transformation (i.e. relativity of simultaneity): The station master's clocks at which the train conductor passes by are all advanced compared to his own because of <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vx/c^{2}\right)}</math>, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation, leading to the effect effect of clock desynchronization by <math>{\scriptstyle t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t+vx/c^{2}\right)-\frac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}}</math>, where the last term is a constant in case the conductor's clock is used as standard for the new synchronization. All of this leads to the result that the conductor's clock is retarded with respect to the station clock at reunion. {{Lorentzbox|Text=Langevin's constant <math>\tfrac{2\theta v^{2}/c^{2}}{\sqrt{1-v^{2}/c^{2}}}</math> corresponds to Thirring's constant <math>t'_{0}</math> .}} |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu3 /> 1923 |{{anchor|Becquerel 1923-DS}}Pierre stays on Earth in the unprimed frame, while Paul makes a round-trip journey in a rocket (in the primed frame) along a clock row resting and synchronized in the Earth frame. At the outbound journey from A to B, Paul notices that Pierre's clocks at which he passes by are all advanced by <math>t=\gamma t'</math> compared to his own clock. He derived this from the temporal Lorentz transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> under the assumption that Paul's clock rests at <math>x'=0</math> in its rest frame S'. After 2 hours Paul reaches B (at which location one of Pierre's clocks indicate 4 hours), and after turnaround he notices again that Pierre's clocks change their synchronization. So one either has to reset and re-synchronize all clocks to zero, or one formally adds a constant time of 2 hours to Paul's clock and 4 hours to all Earth clocks, either way Pierre's clocks at which Paul passes by are more advanced with respect to Paul's. At reunion, Paul confirms Pierre's observation: Paul's clock only indicates 4 hours while Pierre's clock indicates 8 hours. {{Lorentzbox|Text=While Thirring and Langevin used coordinates measured in the ''Earth frame'' S and obtained the coordinates in S' from the transformation <math>t'=\gamma\left(t-vx/c^{2}\right)</math> before turnaround and by changing the sign of v after turnaround, Becquerel instead used coordinates measured in the ''rocket frame'' S' and obtained the coordinates in S from the inverse transformation <math>t=\gamma\left(t'+vx'/c^{2}\right)</math> and by changing the sign of v after turnaround.}} |} ==Curved spacetime== While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.<ref name=koks group=S /> {| class="wikitable" style="background-color:white;{{text default color}};" |- ! Author !! Early examples |- |[[w:Jean Becquerel|Becquerel]]<ref name=becqu1 /> 1922 |After defining gravitational time dilation <math>d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt</math> in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time <math>t</math>. Let us transport clock A to a point where the field is more intense, at a distance <math>r</math> from the center; this clock will measure time <math>\int d\tau</math> which is shorter than <math>\int dt</math>, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B. |} ==Aether interpretations== [[w:Lorentz ether theory]] (LET) is empirically equivalent to special relativity since both models use the Lorentz transformation, thus the description of the twin paradox gives the same result as well. However, special relativity is preferred by the scientific community, because of the validity of the relativity principle in LET is only apparent due to a conspiracy of effects which makes the aether undetecable, and because the measured times and lengths in the aether are supposedly be giving the "true" values even though it is empirically impossible to determine which ones are true or apparent. {| class=wikitable style="background-color:white;{{text default color}};" |- ! style="width:150px" | Author !! Description |- |Langevin (1911)<ref name=langevin1 /> |He argued that it's true that uniform translation with respect to the aether has no experimental meaning in special relativity, though one can still speak of "absolute acceleration" with respect to an aether which he defined as the light carrying medium, so it was "premature to abandon the aether". The absolute nature of acceleration can be seen in the radiation of accelerated charges, as well as in the twin paradox in which the traveler that is more agitated or accelerated is younger at reunion. {{Lorentzbox|Text=Langevin didn't mention this aether interpretation in his later discussions of the twin paradox in 1911/12<ref name=langevin2 /> and 1919.<ref name=langevin3 />}} |- |style="vertical-align:top" |{{Anchor|Wiechert 1911-AE}}Wiechert (1911)<ref name=wiechert11 /> |Contrary to Langevin's emphasis on accelerations, Wiechert emphasized that differential aging can solely be attributed to the existence of ''absolute velocities'' which he called "strides" (German: Schreitungen, denoting the "absolute meaning" of non-accelerated motions) with respect to the aether. In other words, the clock's "strides" are not equivalent or anisotropic with respect to the aether, in contradiction to the "unconditional" relativity principle Einstein's which requires equivalence and symmetry. Wiechert proposed a "conditional" relativity principle in which the aether (in the sense of Lorentz) determines all processes in nature, thus the rate of clocks or processes is "really" altered during motion. {{Lorentzbox|Text=a) This aether interpretation was maintained by Wiechert also in subsequent publications in 1915-1922.<ref name=wiechert15 /><ref name=wiechert20 /><ref name=wiechert21 /> b) Wiechert's interpretation in terms of absolute non-accelerated motions was also independently given by others: [[s:fr:Georges Lechalas]] (1912/13)<ref name=lech /> argued that time dilation and asymmetric aging are the result of absolute speeds with respect to an aether, while criticizing Langevin's (1911) argument that only accelerations can indicate the aether and are responsible for the asymmetric aging; [[w:Edward Vermilye Huntington]] (1912)<ref name=hunt /> argued that the "famous paradoxes of the theory of relativity" represented by the reciprocal nature of length contraction and time dilation, appear as "necessary consequences of perfectly natural and reasonable conventions for setting clocks and laying out coordinates", when the Lorentz transformation is derived and interpreted on the basis of a stationary aether in the sense of Lorentz. c) Laue (1911/12)<ref name=laue1 /> applauded Wiechert for the "particularly striking way" in which he demonstrated that asymmetric aging indeed follows from the relativity laws, yet he rejected Wiechert's aether interpretation and Wiechert's claim that asymmetric aging contradicts the "unconditional" relativity principle: Laue pointed out that the straight worldline connecting two events A and B is only "preferred" insofar as it indicates the maximal proper time between them, yet this preference is ''not'' the result of the natural laws employed, but is rather due to the choice of points A and B; by analogy, the geometric result that two points determine a straight line and thus a direction as well, doesn't refute the physical equivalence of all directions in space; finally, as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.}} |} ==Paradoxical?== Even though Einstein (1905)<ref name=einstein05 /> denoted the round-trip experiment as "peculiar" (German: eigentümlich) he saw nothing contradictory in it. Without mentioning the round-trip experiment, [[w:Paul Gruner]] in May 1910<ref name=gruner10 /> denoted the effect that clocks in relative motion are not showing the same time when they meet as the "great paradox" of relativity (German: Grosse Paradoxie), and [[w:Max von Laue]] in May 1911<ref name=laue0 /> wrote that some consequences of the relativity of time occasionally indeed look "paradoxical" (German: paradox). Eventually, Laue in November 1911 (published 1912)<ref name=laue1 /> was the ''first to specifically denote the round-trip experiment'' as a "paradox" (German: Paradoxie): {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" | German original of [[w:Max von Laue|Laue]] (1911/12):<ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> ! English translation |- |Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist. |Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects. |- |colspan=2|{{Lorentzbox|Text=Subsequently, [[w:Paul Gruner]] (1912)<ref name=gruner /> and others including Einstein (1918)<ref name=einstein18 /> explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger]] (1913)<ref name=seel /> spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").}} |} ==Misunderstandings== {| class=wikitable style="background-color:white;{{text default color}};" ! style="width:50%" padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):<ref name=berg /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |Im Punkte <math>x = 0</math> des Systems S befinde sich eine Uhr, eine andere im Punkte <math>x'=0</math> von S'. Diese zweite bewege sich mit S' bis zum Punkte <math>x = a</math>, kehre dort um und bewege sich nun mit der Geschwindigkeit <math>v</math> zurück bis zum Punkte <math>x= 0</math>. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in <math>x' = 0</math> hat sich mit der Geschwindigkeit <math>v</math> bis zum Punkte <math>x = a</math> bewegt; dazu brauchte sie die Zeit <math>\tau=\tfrac{a}{v}</math>. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit <math>2\tau=2\tfrac{a}{v}</math> ist die Uhr also wieder im Punkte <math>x = 0</math> angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch <math>2\tau=2\tfrac{a}{v}</math>. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist. | style="padding: 0px 20px 0px 20px;" |There is a clock at point <math>x=0</math> of system S, and another one at point <math>x'=0</math> of S'. The second one moves together with S' until point <math>x=a</math>, turns around and now moves back with speed <math>v</math> to point <math>x=0</math>. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at <math>x=0</math> has been moving with speed <math>v</math> until point <math>x=a</math>, for which it required time <math>\tau=\tfrac{a}{v}</math>. The same time is required for the way back. After time <math>2\tau=2\tfrac{a}{v}</math> the clock has thus arrived again at point <math>x=0</math>. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by <math>2\tau=2\tfrac{a}{v}</math> in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free. |- |colspan=2|{{Lorentzbox|Text=a) Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem. b) Berg's mistake also seems to be based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" padding=10 | German original by [[w:Otto Lehmann (physicist)|Lehmann]] (1910/11):<ref name=lehm /> ! English translation |- | style="padding: 0px 20px 0px 20px;" | Hat man zwei genau gleich gehende Uhren nebeneinander und bewegt nun die eine auf einer beliebigen Kurve bis wieder zur anderen zurück, welche Bewegung <math>t</math> Sekunden gedauert haben möge, so geht sie gegen diese um <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> oder annähernd <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> Sekunden zu spät, weil sie während der Bewegung auf langsameren Gang reguliert werden mußte, um mit der ruhenden Uhr in Übereinstimmung zu bleiben. | style="padding: 0px 20px 0px 20px;" | If there are two exactly identical clocks next to each other, and if one of them will be moved on an arbitrary curve until it comes back, which may have lasted <math>t</math> seconds, it [the moving clock] will lag behind the other one by <math>\left(1-\sqrt{1-\tfrac{v^{2}}{c^{2}}}\right)t</math> or approximately <math>\tfrac{1}{2}t(^{v}/_{c})^{2}</math> seconds, because it [the moving clock] had to be regulated to a slower rate during motion in order to remain synchronous with the resting clock. |- |colspan=2|{{Lorentzbox|Text=a) So Lehmann erroneously thinks that the time difference at reunion is only the result of "regulating" the moving clock. This implies that he (like Berg) believed that special relativity predicts ''equal'' clock times at reunion if the moving clock would have been left ''unregulated'' during the round-trip. b) Lehmann's mistake (similar to Berg) seems to based on the fact, that in his paper he generally described time dilation and length contraction of special relativity as only being "apparent" as opposed to Lorentz's theory.}} |- ! style="width:50%" | German original by [[w:Emil Wiechert|Wiechert]] (1911)<ref name=wiechert11 /> ! English translation |- |colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he (like Berg and Lehmann) claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic (compare with section {{slink||Aether interpretations}}). This led him to the following interpretation of the clock paradox: |- | style="padding: 0px 20px 0px 20px;" |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis <math>1:\sqrt{1-u^{2}/c^{2}}</math> zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, <math>+u</math> und <math>-u</math>, nacheinander inne hat. Wird dann zu A der Wert <math>u_{1}</math>, zu B der Wert <math>u_{2}</math>, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen <math>+u_{1}</math>, <math>-u_{1}</math>, oder <math>+u_{2}</math>, <math>-u_{2}</math> weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.'' | style="padding: 0px 20px 0px 20px;" | [...] thus B's progress must be retarded with respect to A's in the ratio <math>1:\sqrt{1-u^{2}/c^{2}}</math> at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides <math>+u</math> and <math>-u</math>. If we apply the value <math>u_{1}</math> to A and <math>u_{2}</math> to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides <math>+u_{2},-u_{2}</math> or <math>+u_{1},-u_{1}</math> are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.'' |- |colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.<ref name=laue1 />}} |- ! style="width:50%" | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)<ref name=camp /> ! English translation |- |colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: |- | style="padding: 0px 20px 0px 20px;" |Dieser Schluß ist nicht richtig. Die Beziehung zwischen <math>t</math>, der Ablesung an der Uhr auf A seitens des Beobachters auf A und <math>t'</math>, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs <math>t=t'</math> ist) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. Der Unterschied zwischen <math>t'</math> and <math>t</math> ist eine Funktion von <math>z</math> und <math>v</math> allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen <math>t'</math> and <math>t</math> wieder auf null zurück, gleichviel, welche Werte <math>z</math> und <math>v</math> während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von <math>v</math> eine endliche plötzliche Änderung. | style="padding: 0px 20px 0px 20px;" |This conclusion is not correct. The relationship between <math>t</math> as the reading on the clock on A by the observer on A, and <math>t'</math> as the reading on the clock on B by the observer on A, is given by (assuming that <math>t=t'</math> at the beginning of the experiment) :<math>t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)</math>. The difference between <math>t'</math> and <math>t</math> is a function of <math>z</math> and <math>v</math> alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between <math>t'</math> and <math>t</math> goes back to zero, no matter what values <math>z</math> and <math>v</math> may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of <math>t'</math> also undergoes a finite sudden change. |- |colspan=2|{{Lorentzbox|Text=a) So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: He is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.) b) As Campbell's paper was a reply to Wiechert, it's interesting to compare them: Campbell was correct in dismissing the aether but wrong in denying asymmetric aging, while Wiechert was correct in deriving asymmetric aging but wrong in requiring the aether. }} |- ! style="width:50%" | French original by [[w:Paul Gruner|Gruner]] (March 1912):<ref name=gruner /> ! English translation |- | style="padding: 0px 20px 0px 20px;" |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable. | style="padding: 0px 20px 0px 20px;" | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction. |- |colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}} |} ==Einstein's contributions== 1905:<ref name=einstein05 /> Introduction of the "peculiar" (German: eigentümlich) round-trip experiment with clocks in terms of a polygonal path as well as a continuously curved path, and an experiment comparing a clock at the pole with one at the equator. January 1911 (published November):<ref name=einstein11a /> In a lecture from January, Einstein extended the "funny" (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911<ref name=lammel /> and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,<ref name=muller /> Einstein spoke of human beings as well. January 1911 (published January 1912):<ref name=einstein3 /> During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because "clocks are carriers of the time differential"; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 1912:<ref name=einst12manu /> In an unpublished manuscript on special relativity, Einstein showed that if system <math>\Sigma'</math> makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system <math>\Sigma </math> at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. April 1914:<ref name=einstpetz /> [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a "fallback into absolutist way of thinking", claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it "must be concluded" that the clock traveling on a polygonal path is retarded at reunion. May 1914:<ref name=rowe group=S /> During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute in the theory of relativity"). 1916:<ref name=einstein16 /> In a lecture of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of general relativity." September 1918:<ref name=einadl /> In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of Berg and Petzoldt. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that "Berg and Petzoldt were wrong". November 1918:<ref name=einstein18 /> In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the "critic" said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the "relativist" who regretfully noted that even some pro-relativity authors tried to "circumvent this unavoidable result". Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem "rises again from the dead" in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is "the double" of its velocity time dilation in the inertial periods 2) and 4). December 1918:<ref name=einstein18b /> In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor <math>1+\Phi/c^{2}</math> in K' in order to show that U1 at distance <math>l</math> is advancing by <math>\Phi/c^{2}=2vl/c^{2}</math> in period 3), which is indeed the double of approximated delay <math>vl/c^{2}</math> caused by velocity time dilation during periods 2) and 4). 1920:<ref name=einstein20 /> In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our "common sense" is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind. August 1920:<ref name=rowe group=S /> At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;<ref name=einst20 /> after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the "well known example of the clocks (or twins)", remarking that the charge of solipsism will be "greeted by the experts as a joke", and characterized the claim that relativity requires mutual retardation of two co-located clocks as a "deliberate attempt to misinform the lay public". 1922:<ref name=morand /><ref name=nord /> [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation. 1954:<ref name=einstein54 /> In a letter, Einstein explained the "well known clock paradoxon" using two clocks <math>B_{1}</math> and <math>B_{2}</math>; clock <math>B_{2}</math> has to reverse its speed in order to come back, thus it was initially at rest in inertial frame <math>S_{2}</math> and then at rest in <math>S_{2}^{+}</math>, whereas <math>B_{1}</math> constantly remains at rest in <math>S_{1}</math>, which explains the asymmetry between them. ==Historical references== <references> <ref name=einstein05>See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].</ref> <ref name=einstein11a>See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.<br /> The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].</ref> <ref name=einstein3>Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu "Die Relativitäts-Theorie"|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}<br /> While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. </ref> <ref name=einst12manu>See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstlor>{{Citation|author=Einstein, A.|date=1914|title=Review of "Lorentz, H. A. – Das Relativitätsprinzip" |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}</ref> <ref name=einstpetz>{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein16>See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture "Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie"|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}</ref> <ref name=einadl>Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}</ref> <ref name=einstein18>Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=[[s:de:Dialog über Einwände gegen die Relativitätstheorie|Dialog über Einwände gegen die Relativitätstheorie]]|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.</ref> <ref name=einstein18b>Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}</ref> <ref name=einstein20>Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206</ref> <ref name=einst20>{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in "The Collected Papers of Albert Einstein", Vol. 7, Document 45}}</ref> <ref name=einstein54>Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])</ref> <ref name=morand>Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}</ref> <ref name=lammel>{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]</ref> <ref name=lammel2>See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}</ref> <ref name=langevin1>He derived differential aging from the proper time integral; pointed out that this demonstrates the "absolute nature of acceleration" with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource</ref> <ref name=langevin2>See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}</ref> <ref name=langevin3>See p. 622f in: {{citation |author=Langevin, P. |title=Le Principe de relativité |journal=Bulletin de la société française des électriciens |volume=9 |date=1919 |pages=601-639|url=https://archive.org/details/bulletin-de-la-societe-francaise-des-electriciens-ser.-3-vol-9/page/600}}; Republished in 1922 as separate booklet: [[:s:fr:Le Principe de relativité|Le Principe de relativité]], Éditions Étienne Chiron</ref> <ref name=wiechert11>See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}</ref> <ref name=wiechert15>See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}</ref> <ref name=wiechert20>See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}</ref> <ref name=wiechert21>See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}</ref> <ref name=muller>See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&issuepage=11 Part 2 published 23 October 1911]}}</ref> <ref name=gruner10>See p. 89 in: {{Citation |author=Gruner, P. |title=Elementare Darlegung der Relativitätstheorie |journal=Mitteilungen der Naturforschenden Gesellschaft Bern |issue=1751 |pages=82-103 |date=May 1912|doi=10.5169/seals-319208}}</ref> <ref name=gruner>See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}</ref> <ref name=weyl>See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.</ref> <ref name=gbaum>See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}</ref> <ref name=laue1>Laue introduces the word "paradox", alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource</ref> <ref name=laue2>See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource</ref> <ref name=laue3>See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource</ref> <ref name=laue0>See preface in: {{Citation |author=Laue, M. v. |date=May 1911|title=Das Relativitätsprinzip (First Edition) |publisher=Vieweg |place=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}</ref> <ref name=berg>See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}</ref> <ref name=camp>See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as "Normann".</ref> <ref name=seel>{{Citation|author=Seeliger, R.|title=Review of "P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther"|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&pg=PA336}}</ref> <ref name=study>See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} </ref> <ref name=robb1>See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} </ref> <ref name=robb2>See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}</ref> <ref name=edding2>See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}</ref> <ref name=rogers>{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}</ref> <ref name=lorentz1>See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource</ref> <ref name=lorentz3>See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}</ref> <ref name=bloch>See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}</ref> <ref name=bloch2>See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}</ref> <ref name=bollert1>See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}</ref> <ref name=born>See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]</ref> <ref name=pauli>See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]</ref> <ref name=thirring>See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}</ref> <ref name=sommerfeld>See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's "Space and Time"|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}</ref> <ref name=kopff>See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].</ref> <ref name=becqu1>See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu2>See p. 57ff (proper time), p. 177f (general relativity), in: {{citation |author=Becquerel, J.|title=[[s:fr:Exposé élémentaire de la théorie d'Einstein et de sa généralisation|Exposé élémentaire de la théorie d'Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}</ref> <ref name=becqu3>Letter from Becquerel to Bergson published by Bergson in 1923, in which Becquerel is not called by name but is referred to as "un physicien des plus distingués", see [[s:fr:Durée et simultanéité/Appendice I|Appendix 1 of Bergson's "Durée et simultanéité"]] on pp. 247ff.</ref> <ref name=nord>Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}</ref> <ref name=lehm>{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip der neue Fundamentalsatz der Physik |title-link=s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}}</ref> <ref name=lech>See p. 19ff in: {{citation |author=Lechalas, G. |orig-date=Submitted December 1912|date=1913 |title=Le nouveau temps |journal=L’Année philosophique|volume=XXIII|pages=19-44|url=https://gallica.bnf.fr/ark:/12148/bpt6k255882z/f22.item}}</ref> <ref name=hunt>See pp. 494ff in: {{Citation|author=Huntington, E. V. |year=1912 |title=A new approach to the theory of relativity|journal=Philosophical Magazine |volume=23|pages=494-512|url=https://archive.org/details/londonedinburg6231912lond/page/494/mode/2up}}</ref> </references> ==Secondary sources== <references group=S> <ref name=miller>{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)</ref> <ref name=lange>{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}</ref> <ref name=pes>{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}</ref> <ref name=during>{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).</ref> <ref name=debs>{{Citation |author=Debs, T. A., & Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}</ref> <ref name=alizzi>{{Citation |author=Alizzi, A., Sen, A., & Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury</ref> <ref name=beng>{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.</ref> <ref name=rowe>{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.</ref> <ref name=weiss>Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]</ref> <ref name=cuvaj>{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}</ref> <ref name=koks>Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]</ref> </references> [[Category:History of special relativity]] [[Category:Paradoxes]] edf98i32eqiw82qf64nd84vcj2kk0bt 2 322414 2835016 2761072 2026-09-29T09:40:34Z Dr rajatsubhra 188238 Updated RMDMCC protocol and preliminary clinical outcomes to align with the 2026 preprint; added current Zenodo DOI 10.5281/zenodo.23030017, updated safety/limitations, and removed obsolete Zenodo reference. 2835016 wikitext text/x-wiki == RMDMCC – A Non-Surgical Dilatation Method for Low-set Hirschsprung’s Disease == RMDMCC describes a clinician-supervised, caregiver-performed, non-surgical approach for selected cases of '''low-set Hirschsprung’s Disease''', using progressive manual dilatation, supportive bowel management, and close clinical follow-up. Surgery remains the established definitive treatment for Hirschsprung’s disease. == Clinical Background == '''Hirschsprung’s Disease (HD)''' is a congenital disorder characterized by the absence of enteric ganglion cells in the affected bowel segment, resulting in functional obstruction, chronic constipation, abdominal distension, and related bowel dysfunction. Standard definitive treatment involves surgical resection of the affected aganglionic segment. RMDMCC has been explored as a conservative approach in selected children clinically and radiologically identified as having low-set disease. The available RMDMCC evidence remains preliminary and observational. == About RMDMCC == '''RMDMCC (Rajatsubhra’s Manual Dilatation Method for Congenital Constipation)''' is a clinician-supervised, caregiver-performed progressive manual dilatation protocol developed for selected children with low-set Hirschsprung’s disease. The protocol emphasizes progressive manual anal–rectal dilatation together with supportive bowel management and clinical follow-up, with the aim of improving bowel evacuation in appropriately selected children. == Practical Protocol (Clinician-supervised, Caregiver-performed) == === Prerequisites and Setup === * Identify selected children with suspected or clinically/radiologically diagnosed '''low-set or ultrashort-segment Hirschsprung’s disease''' or congenital distal outlet obstruction. * Exclude long-segment or total colonic Hirschsprung’s disease and consider other causes of childhood constipation. [#1][#2] * Obtain written informed consent and document caregiver training. * Explain warning signs requiring prompt reassessment, including fever, vomiting, increasing abdominal distension, bleeding, significant pain, poor feeding, poor growth, or symptoms suggestive of enterocolitis. === Hygiene and Preparation === * Trim nails and wash hands thoroughly. * Use non-powdered nitrile or latex gloves. * Use an appropriate water-based lubricant. * If topical local anaesthetic is considered necessary, it should be used cautiously and according to appropriate pediatric clinical guidance. === Technique (Stepwise Manual Finger Dilatation) === * Frequency: '''4–5 sessions per day''', approximately '''5 minutes each'''. * The procedure should be performed hygienically and according to the training provided by the supervising clinician. # Begin with the '''little finger''', adequately lubricated and gloved. # Insert gently following the natural anorectal direction without force. # If impacted stool is encountered, allow evacuation and reassess before continuing. # Identify the constricted distal anal or rectal segment clinically. # Apply only gentle, controlled pressure and avoid forceful manipulation. # Progressively increase finger thickness according to the child's tolerance. # Each stage of progression generally continues for approximately '''4–6 days''', depending on tolerance. # Progress from the '''little finger''' to progressively thicker fingers. # When comfortably tolerated, progress to the '''index and middle fingers one over another'''. # Subsequently progress to '''two fingers side-by-side'''. # In the final stage, the two fingers may be gently widened or separated as much as can be comfortably accommodated within the constricted zone, without force. # The final stage is generally reached over approximately '''1–2 months''', although progression varies according to individual tolerance. # Continue the overall protocol, with clinical supervision and bowel-management support as required, for up to approximately '''6 months'''. # Stop the procedure and seek clinical reassessment if pain, bleeding, significant resistance, worsening symptoms, or poor tolerance occurs. The procedure should remain under the supervision of a pediatrician or appropriately trained clinician experienced in pediatric bowel management. === Laxative and Supportive Measures === * '''Polyethylene Glycol (PEG)''' or '''Lactulose/Lactitol''' may be used as supportive therapy and titrated according to stool pattern and clinical response. [#15] * Maintain adequate hydration and age-appropriate nutritional support. * Record stool frequency, stool consistency, abdominal symptoms, discomfort, and other clinically relevant observations where feasible. * Stool-softener requirements should be reassessed according to bowel pattern and clinical response rather than discontinued at a predetermined fixed time. During follow-up, some caregivers reported that their children developed regular bowel movements without continued stool-softener requirement. This remains a caregiver-reported clinical observation and has not been established as a controlled treatment effect. === Follow-up and Timeline === * Review clinically at regular intervals, commonly every 2–4 weeks during the initial period. * Progression of dilatation should depend on the individual child's tolerance. * The final dilatation stage may be reached over approximately '''1–2 months'''. * Clinical and bowel-management follow-up may continue for up to approximately '''6 months''' or according to individual clinical requirements. * Follow-up '''Barium Enema''' may be considered at approximately 6 months to assess radiological changes where clinically appropriate. [#20] * Reduction or discontinuation of stool softeners should be individualized according to bowel pattern and clinical assessment. === Safety === The procedure should not be forceful. Clinical reassessment is required if the child develops: * Pain or significant distress * Bleeding * Increasing resistance to dilatation * Increasing abdominal distension * Vomiting * Fever * Poor feeding or poor growth * Deterioration in bowel function * Suspected Hirschsprung-associated enterocolitis Failure to improve or clinical deterioration should prompt specialist pediatric surgical or pediatric gastroenterology evaluation. === Escalation and Specialist Input === * Persistent symptoms, recurrent enterocolitis, poor growth, worsening abdominal distension, or failure of conservative management should prompt pediatric surgical or gastroenterology assessment. * Other established bowel-management interventions may be considered by appropriate specialists according to the clinical situation. [#5][#6] * Definitive surgical management should not be delayed when clinically indicated. === Documentation and Ethics === Clinical documentation should include, where applicable: * Signed guardian consent. * Caregiver training. * Clinical diagnosis and available investigations. * Stool-pattern monitoring. * Growth and general clinical monitoring. * Treatment progression. * Adverse events. * Need for laxatives or other bowel-management measures. * Follow-up findings. * Surgical referral where required. RMDMCC was developed and clinically practiced at '''Child Health Care Arambag (CHCA)''', West Bengal, with institutional support from '''Sri Yoga Center Trust – Kunarpur'''. == Key Clinical Rationale == * Digital finger-based dilatation provides tactile feedback during gradual progression and allows progression to be adjusted according to patient tolerance. * The protocol is intended to facilitate bowel evacuation through gradual dilatation of the constricted distal segment. * Progression is gradual rather than forceful. * Supportive bowel management remains an integral part of the protocol. * Preliminary clinical observations suggest improvement in bowel regularity in selected cases, but these observations require confirmation through prospective standardized studies. == Preliminary Clinical Outcomes == The currently documented RMDMCC experience includes '''32 children''' clinically and radiologically identified as having low-set Hirschsprung’s disease. Descriptive observations from this preliminary case series include: * '''30/32 (93.75%)''' were reported to have achieved symptomatic resolution without surgery during the documented observation period. * '''28/32 (87.5%)''' showed reported radiological improvement. * '''2/32''' were referred for surgical management. * Some caregivers reported regular spontaneous bowel movements without continued stool-softener requirement during follow-up. * Historical caregiver documentation includes favorable reports as well as at least one report of no benefit followed by treatment elsewhere. These findings are '''preliminary and observational'''. The case series was uncontrolled, diagnosis was largely based on clinical and radiological assessment, and histopathological confirmation was not documented for every case. Some outcomes were based partly on caregiver reports, and standardized long-term follow-up was incomplete. Therefore, these descriptive findings should not be interpreted as proof that RMDMCC is equivalent or superior to established surgical treatment. The detailed 2026 clinical-method and preliminary case-series documentation is available as a Zenodo preprint: https://doi.org/10.5281/zenodo.23030017 ''An Innovative Pediatric Protocol developed by Dr. Rajatsubhra Mukhopadhyay'' == Overview == RMDMCC (Rajatsubhra’s Manual Dilatation Method for Congenital Constipation) is a non-surgical clinical method developed by '''Dr. Rajatsubhra Mukhopadhyay, MBBS, DCH, MD''' for selected children with low-set Hirschsprung’s disease. Dr. Mukhopadhyay serves as Director and Principal Investigator of '''Child Health Care Arambag (CHCA)''', West Bengal, a Clinical & Research Wing supported by '''Sri Yoga Center Trust – Kunarpur'''. == How RMDMCC Differs == '''RMDMCC (Rajatsubhra’s Manual Dilatation Method for Congenital Constipation)''' uses a structured, outpatient-based, '''progressive manual anal–rectal dilatation technique''' in selected children with low-set disease. Unlike an instrumental dilatation protocol, RMDMCC uses progressive finger-guided dilatation, with the degree of progression determined by patient tolerance and clinical supervision. === Distinctive features === * Uses progressive finger-guided dilatation over time rather than forceful progression. * Frequency is generally '''4–5 sessions daily''' during the active dilatation period. * Each progression stage generally lasts approximately '''4–6 days''', depending on tolerance. * Progresses from the little finger through thicker fingers, followed by two-finger techniques when tolerated. * The final stage is generally reached over approximately '''1–2 months'''. * Overall clinical management and follow-up may continue for approximately '''6 months'''. * Incorporates supportive bowel management, including osmotic stool softeners when clinically required. * Requires caregiver training, informed consent, clinical monitoring, and reassessment for adverse symptoms. * Preliminary observational experience documents non-surgical symptom resolution in a proportion of selected cases, while recognizing that surgery remains the established definitive treatment for Hirschsprung’s disease. === Ethical and Safety Framework === * Appropriate clinical assessment is required before initiating the protocol. * Long-segment or total colonic disease and other important causes of constipation should be considered. * The procedure should not be forceful. * Pain, bleeding, significant resistance, clinical deterioration, suspected enterocolitis, or failure to improve requires reassessment. * RMDMCC is '''not a substitute for definitive pull-through surgery when surgery is clinically indicated'''. * Patient guardians should provide informed consent after explanation of the procedure, alternatives, uncertainties, and need for follow-up. == Academic Relevance == RMDMCC represents a clinical method requiring further systematic investigation. Potential areas of academic interest include: * Pediatric bowel management * Low-set Hirschsprung’s disease * Conservative management strategies * Resource-limited pediatric care * Caregiver-performed interventions under clinical supervision * Prospective evaluation of non-surgical clinical methods The currently available RMDMCC evidence should be interpreted as preliminary clinical documentation rather than definitive evidence of efficacy. == Future Scope == Future evaluation should include: * Prospective standardized case documentation * Clearly defined diagnostic criteria * Standardized clinical outcome measures * Systematic adverse-event monitoring * Quantification of stool-softener requirements * Longer-term follow-up * Independent or multicentric evaluation * Comparison with established management pathways * Submission of appropriately designed studies to peer-reviewed pediatric or pediatric-surgical journals Prospective studies using standardized diagnostic criteria, predefined outcomes, adverse-event monitoring, and longer-term follow-up are needed to evaluate the safety and clinical utility of RMDMCC more rigorously. == References == # Ambartsumyan L, et al. ''Evaluation and Management of the Postsurgical Hirschsprung Patient''. NASPGHAN; 2023. https://naspghan.org/wp-content/uploads/2023/03/Evaluation_and_Management_of_Postsurgical_Patient.27-4.pdf # Lindert J, et al. ''Bowel Management in Hirschsprung Disease—Pre-, Peri- and Postoperative Aspects''. Children (Basel). 2024;11(3):371. https://doi.org/10.3390/children11030371 # Tabbers MM, et al. ''Evaluation and Treatment of Functional Constipation in Infants and Children: Evidence-Based Recommendations''. JPGN. 2014;58:258–274. # Gordon M, et al. ''ESPGHAN and NASPGHAN protocol for paediatric functional constipation treatment guidelines''. BMJ Paediatr Open. 2025;9:e003161. # Mosiello G, et al. ''Consensus Review of Best Practice of Transanal Irrigation in Children''. JPGN. 2017;64(1):38–46. # Baaleman DF, et al. ''Transanal Irrigation in Children: Treatment Success and Health-related Quality of Life''. J Pediatr Gastroenterol Nutr. 2022;74(6):847–853. # Bokova E, et al. ''State of the Art Bowel Management for Pediatric Colorectal Problems''. Children (Basel). 2023;10(7):1178. # Roorda D, et al. ''Botulinum toxin injections after surgery for Hirschsprung disease: Systematic review and meta-analysis''. J Pediatr Surg. 2019;54(11):2419–2426. # Rentea RM, et al. ''Impact of Botulinum Toxin on Hirschsprung-Associated Enterocolitis and Obstructive Symptoms''. J Pediatr Surg. 2021;56(7):1201–1206. # Halleran DR, et al. ''Anal sphincter botulinum toxin injection in children with Hirschsprung disease and severe functional constipation''. J Pediatr Surg. 2019;54(11):2430–2435. # Tatekawa Y. ''A case of ultrashort-segment Hirschsprung’s disease''. J Surg Case Rep. 2021;2021(8):rjab357. # Xie C, et al. ''Clinical features and prognosis: ultrashort-segment vs short-segment HD''. Front Pediatr. 2023;10:1061064. # West of Scotland Neonatal Network. ''Guidelines for the management of bowel irrigation (rectal washout) for infants <1 year''. 2017. # Guideline Central. ''NASPGHAN—Evaluation and Management of Postsurgical Hirschsprung''. 2023. # de Geus A, et al. ''Update of Pharmacological Management in Children with Functional Constipation''. Paediatr Drugs. 2023;25:559–575. == Notes for Readers == * RMDMCC remains a '''preliminary clinical method''' supported by observational documentation rather than controlled evidence. * The reported 32-case experience should not be interpreted as establishing comparative efficacy against surgery or other treatments. * Diagnosis in the documented series was largely clinical and radiological; histopathological confirmation was not documented for every child. * Some outcomes and follow-up observations were caregiver-reported. * For suspected or confirmed Hirschsprung’s disease, appropriate pediatric surgical involvement remains important. * Definitive surgical treatment should not be delayed when clinically indicated. == Sources == # Mukhopadhyay, Rajatsubhra. ''Manual Dilatation for Congenital Constipation – RMDMCC Technique''. Self-published Clinical Manual, 2024. # Mukhopadhyay, Rajatsubhra. ''RMDMCC Clinical Demonstration and Case Compilation''. ResearchGate, 2025. ResearchGate: https://www.researchgate.net/publication/392978649_RMDMCC_A_Novel_Non-Surgical_Approach_for_Low-Set_Hirschsprung's_Disease === Supporting Materials === '''Zenodo – Current 2026 Preprint:''' https://doi.org/10.5281/zenodo.23030017 '''Figshare:''' https://doi.org/10.6084/m9.figshare.29587274 '''GitHub:''' https://github.com/drrajatsubhra/RMDMCC-treatment-method === Citation === '''Current RMDMCC Preprint (2026):''' Mukhopadhyay, R. (2026). ''RMDMCC – A Non-Surgical Dilatation Method for Low-Set Hirschsprung's Disease: Clinical Method and Preliminary Case-Series Documentation''. Zenodo. https://doi.org/10.5281/zenodo.23030017 '''Earlier Supporting Record:''' Mukhopadhyay, R. (2025). ''RMDMCC – A Non-Surgical Protocol for Low-set Hirschsprung’s Disease''. Figshare. https://doi.org/10.6084/m9.figshare.29587274 '''Note:''' The RMDMCC materials cited here, including the 2026 Zenodo preprint, represent preliminary clinical documentation. The reported outcomes are observational and require further prospective evaluation. This page does not present the available documentation as evidence that RMDMCC is equivalent or superior to established definitive surgical treatment. == Licensing == This page is released under the following license: ''Creative Commons Attribution-ShareAlike 3.0'' (CC BY-SA 3.0) You are free to reuse, adapt, and share with attribution. <!-- == Categories == [[Category:Pediatrics]] [[Category:Innovative Medical Methods]] [[Category:Wikiversity Health Projects]] [[Category:India]] [[Category:Alternative Therapies]] --> 4x6n2tyt9jwrnhcqtom36p352yk4uz5 0 323118 2834960 2761762 2026-09-29T03:50:27Z P U3270518 3106535 corrected spelling error 2834960 wikitext text/x-wiki {{title|Inner voice in anxiety:<br>What role does the inner voice play in maintaining or alleviating anxiety?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Cartesian Theater.jpg|thumb|'''Figure 1'''. The sensation that it can feel like theres{{g}} a whole different person inside the head.]] ;Imagine this.... It's finally your turn to present to your class, {{g}} you've spent hours preparing and practicing, everything is ready and this assessment will finally be over. Then suddenly you begin hearing words such as;{{g}} "everyone is staring at you", "don't stutter", "maybe your presentation is completely wrong", etc. That voice is your own inner voice greatly heightening your anxiety, but then you hear that same inner voice say things such as;{{g}} "you are ready", "it's just a few minutes", etc. Then suddenly your anxiety is relieved and you are once again ready to present. This is a very common real-life example of the power of our inner voices regarding levels of anxiety. {{RoundBoxBottom}} The inner voice, otherwise known as self-talk, internal monologue, etc, is the constant voice heard in one’s mind, shaping thoughts, behaviour and emotions (Alderson-Day & Fernyhough, 2015). [[Anxiety]] on the other hand, having uncomfortable situations or thoughts cause a state of negativity, worry and distress. Often accompanied with increased heart rate, muscle tension and other physical reactions (Barlow, 2002){{g}}. There is a strong yet complex relationship between the inner voice and anxiety with worry and rumination often being a result of negative self-talk, yet healthy [[Regulation of emotion|emotional regulation]] also being often the result of constructive and/or reflective self-talk{{g}}. This chapter explores various theories/models regarding this topic, along with a set of empirical findings and [[Meta-analysis|meta-analyses]] to test these theories and ultimately prove just how strong the relationship is between the inner voice and anxiety. {{RoundBoxTop|theme=3}} '''Focus questions''' * In what ways can goal-directed self-talk within the inner voice reduce anxiety levels? * How can the inner voice worsen anxiety levels? * What approaches of inner voice usage provide the most reliable ways to improve everyday anxiety? * What should future research focus on improving regarding inner voice interventions to relieve anxiety levels? {{RoundBoxBottom}} =='''Theory'''== {{expand}} ==Definitions== Inner voice, also known as self-talk, covert speech, internal monologue, automatic thoughts, etc is best defined as the consistent and ever-lasting voice heard in one's mind{{f}}. Inner voice has been proven to have a significant role in the self-regulation of cognition and behavioural decisions/thoughts across the life span (Alderson-Day & Fernyhough, 2015). Anxiety is best defined as a state of negative mood caused by uncomfortable situations or thoughts, often accompanied with physical reactions such as increased heart rate, muscle tension, etc (Barlow, 2002). ==Information Processing Model of Anxiety== Beck & Clark's Information Processing Model of Anxiety is an example of a cognitive model which outlines how the content of the inner voice can relieve or alleviate levels of anxiety{{f}}. This model states that the formation of anxiety involves three main steps. First, the immediate registration of threat, second, activation of the primal threat approach and third, activation of elaborative and reflective thinking. It's in this third step that inner voice becomes vital in the development of anxiety. If this third step is filled with catastrophising, exaggerating, etc, levels of anxiety are likely to increase. However, if this step is filled with more constructive and reflective modes of thinking, levels of anxiety are likely to decrease. Thus, regarding the treatment of anxiety, the goal should be reducing the importance of the activation of the primal threat approach and rather, strengthen the individual’s ability to develop constructive and reflective thinking (Beck & Clark, 1997). ==Attentional Control Theory== Attentional Control Theory is an approach to anxiety, developed from the Processing Efficacy Theory{{f}}. It states that anxiety reduces the efficiency of the goal-directed attentional system and simultaneously increases attention for threat related situations. Two main functions involving attentional control are harmfully affected by anxiety, these are inhibition and shifting. Although, {{g}} this reaction often results in compensatory strategies, allowing the potential for none/minimal performance hindrance (Eysenck, et al, 2007). Attentional Control Theory further makes the assumption that inner verbal activity typically causes more detrimental effects of anxiety rather than imagery for instance{{f}}. This is due to its presence in the phonological loop (the subsystem of working [[memory]]), thus having a higher likelihood of impact on anxiety than other stimulants (Rapee, 1993). Ultimately, Attentional Control Theory predicts that a negative inner voice can interfere with attention levels and therefore risk raising anxiety. ==Interdisciplinary Review and Transdisciplinary Model== A relatively recent interdisciplinary review and transdisciplinary model on self-talk by Latinjak, et al, (2023) analysing 559 self-talk articles resulted in various self-talk interventions across numerous disciplines. This model defines self-talk (inner voice) as the ‘verbalisation directed at the self, able to alter emotions and behaviour’. Furthermore it is stated that self-talk interventions are capable of helping individuals with [[addiction]] problems, mental disorders (anxiety/[[Depressive disorder|depression]]) and promoting an overall healthier lifestyle. The model then states that individuals with a lack of control regarding negative spontaneous self-talk are associated with common symptoms of anxiety including, worry and rumination. In resolving these negative traits, practice and repetition of goal-directed self-talk is successful in improving many psychological skills, specifically, emotional control. Examples of beneficial goal-directed self-talk in an anxiety based context can include constant reminders of past success in similar situations or repetition of positive affirmations (Latinjak, et al, 2023). ==Self-Distancing Theory== Self-Distancing Theory explores personal reflection through a lens that doesn’t involve first-person pronouns{{f}}. The theory best explains this using Malala Yousafzai experiences. Yousafzai, Pakistani Activist and Nobel Peace Prize winner, was famously shot by a Taliban gunman on her way home from school as a 15 year old. When asked how she coped with the stress knowing the possibility of being attacked again Yousafzai replied; “I used to think that the Tali (Taliban) would come and he would just kill me. But then I said, if he comes, what would you do Malala? Then I would reply to myself, Malala just take a shoe and hit him” Yousafzai demonstrates here how to use your inner voice with a third-person perspective, stating that it was beneficial in managing her stress. This example of self-distancing demonstrates a strategy to essentially reinterpret threat and aid emotional control. To conclude, the Self-Distancing Theory predicts that usage of third-person within one’s inner voice helps relieve anxiety. == '''Research''' == {{expand}} ==Catastrophizing== Catastrophizing is the process of repetitive exaggeration of possible negative outcomes, often in one's inner voice. Examples of inner voice catastrophizing can include repeating phrases such as; “What if I never get better”, "Everyone's going to be staring at me”, “This is so unfair”, etc. It is likely that most individuals at some point in their life have said these, and/or similar phrases, specifically through their inner voice if in public or around others. Research proves that this process often has a strong relationship with mental disorders, specifically anxiety (Petrini & Arendt-Nielsen, 2020). Harte, et al (2024), completed a meta-analysis on 57 studies (N=12,603) among young adults/adolescents with chronic pain, analysing and evaluating any correlations between catastrophizing and anxiety, depression, disabilities and pain intensity. Results displayed that catastrophizing had a stronger correlation with anxiety and intensity, notably higher in samples with more males (r=0.50). Additionally, the results show a moderate heterogeneity (I^2=42.32), proving meaningful variability (Harte, et al, 2024){{ic|Not in the References}}, as seen in ''Figure 2''. Ultimately, this meta analysis proves that negative exaggerations, or catastrophizing increases anxiety regarding chronic pain. [[File:Figure 2 harte.png|thumb|589x589px|Figure 2. Table from Harte, et al, results of anxiety symptoms correlation with catastrophizing (result and heterogeneity highlighted)]] Additionally, Pan, et al, (2021), completed an experiment testing the relationship between catastrophizing and anxiety among individuals diagnosed with Somatic Syndrome Disorder (SSD) (N=28). The participants were given the Hamilton anxiety scale (HAMA) and the catastrophizing subscale within the cognitive emotion regulation questionnaire (CERQ) to test this relationship. Results show that the CERQ results were positively and significantly related to the HAMA scores (r=0.416, p=0.049), as seen in ''Figure 3'' (Pan, et al, 2021). Whilst this experiment was targeted towards individuals with SSD, the results remain relevant for the general population who experience catastrophizing and anxiety. [[File:Fig 3 catastrophizing.png|thumb|356x356px|'''Figure 3.''' Table from Pan, et al, results of HAMA results regarding anxiety levels correlation to results of CERQ regarding catastrophising.]] ==Self-Reassurance== Self-reassurance refers to the usage of supportive, [[Empathy|empathetic]], understanding and compassionate speech to one’s self, often within the inner voice. Examples of this include, “It’s ok to fall short”, “I’ll get it next time”, “I did my best and that's all that matters”, etc. Research has proven that practising self-reassurance often helps relieve anxiety. For instance, Barcaccia, et al (2020) applied a stepwise, multiple linear regression to the data of participants with ages ranging from 13-22 (N=424). Participants completed measures regarding a range of mental disorders and coping mechanisms, including anxiety and self-reassurance. Results found that self-reassurance was among the highest protective mechanisms regarding anxious tendencies, explaining 40% of the variance (Barcaccia, et al, 2020). Additionally, Arenella & Steffen (2020) conducted a study investigating the impact of self-reassurance on anxiety for female family caregivers (N=150). Using hierarchical regression models, including the Depression Anxiety and Stress Scale (DASS) and an online screening survey questionnaire results displayed that anxiety was significantly negatively correlated with self-reassurance (r=-0.36, p=0.001) (Arenella & Steffen, 2020). Ultimately, both these studies prove that using self-reassurance is a very effective way to reduce anxiety and/or similar symptoms. However, the most notable limitation of both these studies is the reliance on self-reporting, increasing the risk of bias, meaning that future research should aim to avoid such possibility. ==Self-Distancing== Self-distancing is the process of referring to yourself from another perspective, most commonly by using third-person language. This process may entail using gender-based pronouns or names rather than, “me” or “I”, often within the inner voice. Though this slight change in wording may appear minimal, research has demonstrated that it has an impact on anxiety levels. Kross, et al (2014) analysed seven studies (N=585) testing how different uses of language impacts social stress/anxiety. Overall, the studies display that practicing self-distancing benefits individuals who experience social anxiety, improving their ability to regulate thoughts. Specifically, study three examined participants in a public-speaking scenario, with half performing third-person language and the other half performing non-third-person language. The third-person group displayed less distress and emitted less maladaptive postevent processing (Kross, et al, 2014). Ultimately, this research proves that the language used within the inner voice can play a significant role in emotional regulation, validating personal reports of this effect, such as Yousafzai’s experience previously mentioned. Finally, one common limitation of studies regarding self-distancing’s impact on anxiety is that the findings are often over just a short period of time and are often self-reported. ==Affect Labelling== Affect labelling within the inner voice is essentially admitting to any effects being felt by specifically labelling them, e.g. “I’ve been depressed recently”, “This is making me anxious”, etc. Similarly to self-distancing, this slight change in language may seem minimal, however, numerous studies have proven this process to significantly impact emotional regulation. Neurological evidence displays that affect labelling produced increased activity within the right ventrolateral prefrontal cortex (RVLPFC) and simultaneously reduced the response of the amygdala. This suggests that affect labelling can diminish/reduce emotional reactivity regarding various negative feelings, including anxiety (Lieberman, et al, 2007). Niles, et al (2015) conducted a test on a group of individuals (N=100) who self-reported to experience anxiety when performing [[public speaking]]. The group was split in two, with participants randomly assigned to exposure therapy with either affect labelling or no affect labelling. The results show that the group in the affect labelling exposure therapy displayed a decrease in anxiety by measuring the participants skin conductance and heart rate during their speeches. [[File:Af lab hr 2.png|thumb|555x555px|'''Figure 4.''' Effect of group on heart rate (A) and non-specific skin conductance response (B) during recovery following the speech]] ''Figure 4'' displays the results of each group's heart rate (A) and skin conductance response (B). Firstly, regarding heart rate (''Figure 4A)'', the affect labelling group displayed a relatively stable heart rate from baseline to post exposure, however, a decrease is then noticeable before re-testing. This differs from the non-affect labelling group, proving that the process of affect labeling diminished the expected increase of heart rate after completion of public speaking. Additionally, ''Figure 4B'' displays the results of skin conductance response, similarly showing a significant decrease between post exposure and re-test as heart rate did. These results between post exposure and re-test are; b=1.14, p=0.023, d=1.0, proving that the affect labelling group recovered much better post exposure than the non-affect labelling group (Niles, et al, 2015). Together, this study suggests that affect labelling successfully decreases experiences of anxiety during and post public speaking. Proving that practicing affect labeling within the inner voice can help reduce anxiety in common stressful situations. ==Motivational Self-Talk== [[Motivation|Motivational]] self-talk is the process of saying to yourself inspiring and empowering messages or phrases such as; “I can do it”, “Keep pushing”, “This is easy for me”, etc. This is a very common and powerful tool within one’s inner voice to help boost confidence and limit anxiety-based tendencies under pressure. Hatzigeorgiadis, et al (2009) tested the benefits of motivational self-talk within a sports setting. A group of tennis players (N=72) completed a baseline assessment, three training sessions and finished with a final assessment to conclude the experiment's findings. Following the baseline assessments, the participants were split into two groups, a controlled group and an experimental group. The experimental group completed training in motivational self-talk in each of the three training sessions. To compare results regarding anxiety a two-way mixed model [[MANOVA]] was completed, revealing that the experimental group had an increased level of self-confidence (p=<0.01) and a decreased level of cognitive anxiety (p=<0.05), whereas there were no noticeable changes for the controlled group (Hatzigeorgiadis, et al, 2009). These results suggest that motivational self-talk can not only increase confidence but also reduce anxiety in performative situations, promoting the process of common practice in motivational self-talk within the inner voice. Overall, these empirical findings and [[Meta-analysis|meta-analyses]] conclude that the different mechanisms of the inner voice can impact anxiety within their own way. Self-reassurance, self-distancing, affect labelling and motivational self-talk are proven to effectively reduce anxiety levels when used correctly. However, alternatively, catastrophizing is proven to significantly raise anxiety levels within certain contexts. However, most of these studies focus on short-term impacts and are situationally, meaning future research should aim to target more broad real-word situations to further prove the significance the inner voice has on anxiety. == '''Integration''' == Referring back to Beck & Clark's Information Processing Model of Anxiety (1997), it is stated that anxiety is formed in three main steps, registration of threat, activation of the primal threat approach and activation of elaborative and reflective thinking (Beck & Clark, 1997). This model suggests that anxiety based interventions should focus on strengthening the third step of elaborative and reflective thinking, this is supported by Niles, et al, (2015) study on affect labeling. Affect labelling was used as a tool to reduce anxiety levels when faced with public speaking, with the most notable results appearing post exposure, prior to retesting with skin conductance results most exposure displaying; b=1.14, p=0.023, d=1.0 (Niles, et al, 2015). Essentially, this positive effect of labelling post exposure has acted as a tool to improve the activation of elaborate and reflective thinking as recommended by Beck & Clark's Information Processing Model of Anxiety (1997), thus reducing levels of anxiety. The Attentional Control Theory (Eysenck, et al, 2007) previously stated that anxiety reduces the efficiency of the goal-directed attentional system and increases attention for threat related situations. Additionally, the inner voice verbal activity can cause detrimental effects of anxiety due to its presence in the phonological loop (Rapee, 1993). This theory is supported by research on the inner voice mechanism of catastrophizing, consuming the individual to a level that the goal-directed attentional system becomes unreachable. However, research on motivational-self talk can also integrate to this theory's statement of anxiety increasing attention for threat related situations. The pressure that sporting situations cause for athletes can leave them feeling threatened to perform, Hatzigeorgiadis, et al (2009) tennis player study can be applied here. Due to the participants increase of self-confidence (p=<0.01) and decreased level of cognitive anxiety (p=<0.05), it is proved that these tennis players do normally feel anxious in threat related situations and also that practicing motivational self-talk proves the inner voice can positively impact anxiety. Referring to the interdisciplinary review and transdisciplinary model on self-talk by Latinjak, et al, (2023) it is stated that self-talk (inner voice) is the ‘verbalisation directed at the self, able to alter emotions and behaviour’ (Latinjak, et al, 2023). The model further states that individuals who experience a lack of control regarding negative spontaneous self-talk often possess symptoms of anxiety such as worry and rumination. Latinjak, et al (2023) suggest that goal-directed self-talk can help reduce anxiety, this is supported by both the self-reassurance (Barcaccia, et al, 2020) and the motivational self-talk (Hatzigeorgiadis, et al, 2009) research. Both studies present the benefits of goal-directed self-talk with evidence to ultimately prove Latinjak, et al,  (2023) model. *Self-Distancing Theory is the process of referring to yourself without first-person pronouns, most commonly done within the inner voice using one’s own name or pronouns such as he/she. Research within this theory, including self-reports and testing has proven it to be effective in relieving anxiety. An example of this theory being proven is Kross, et al’s, (2014), previously discussed examination of testing the effectiveness of third-person language for relieving participants anxiety in a public speaking scenario. With results displaying a decreased level of distress and maladaptive postevent processing, integrating perfectly to the theory and thus proving its effectiveness in relieving anxiety levels. =='''Learning feature'''== ;<nowiki>Quiz:</nowiki> <quiz display="simple"> {Research shows that self-distancing (using third-person language) can help reduce anxiety: |type="()"} + True - False {Catastrophizing typically helps reduce anxiety levels: |type="()"} - True + False </quiz> =='''Conclusion'''== Referring to the focus questions in the overview section, firstly, in what ways can goal-directed self-talk within the inner voice reduce anxiety levels? The research regarding motivational self-talk specifically helps answer this question. Hatzigeorgiadis, et al (2009) concluded that a sample of tennis players experienced an increased level of self-confidence (p=<0.01) and a decreased level of cognitive anxiety (p=<0.05), after practicing motivational self-talk (Hatzigeorgiadis, et al, 2009). As the goal was to perform at a higher level under pressure during tennis, this research proves that goal-directed self-talk can reduce anxiety levels. Secondly, how can the inner voice worsen anxiety levels? Models such as Beck & Clark's Information Processing Model of Anxiety (Beck & Clark, 1997) and Attentional Control Theory (Eysenck, et al, 2007) can be used to explain how worry, rumination, catastrophizing, etc has a substantial effect on anxiety. Harte, et al (2024) proves this. With results showing that catastrophizing had a stronger correlation with anxiety and intensity (Harte, et al, 2024). Thus proving catastrophizing is just one way the inner voice can worsen anxiety levels. Thirdly, what approaches of inner voice usage provide the most reliable ways to improve everyday anxiety? Niles, et al, (2015) found that the group that practised affect labelling resulted with; b=1.14, p=0.023, d=1.0 between post exposure and re-test, proving much more effective than the group who did not practice affect labelling (Niles, et al, 2015). Public speaking is a core skill most individuals will have to utilise in some way or another throughout the week. Therefore, incorporating affect labelling is just one way the inner voice can be used to relieve everyday anxiety. Finally, what should future research focus on improving regarding inner voice interventions to relieve anxiety levels? As previously stated, the majority of research and evidence regarding this topic focuses on short-term effects, and can be context-specific. Therefore, any future interventions should aim to help create/test long-term effects whilst being more applicable to everyday situations. ==See also== * [[w:Generalised_Anxiety_Disorder|Generalised Anxiety Disorder]] * [[Stress Management]] =='''References'''== {{Hanging indent|1= Aaron T. Beck, David A. Clark. (1997). An information processing model of anxiety: Automatic and strategic processes, Behaviour Research and Therapy, Volume 35, Issue 1, Pages 49-58, ISSN 0005-7967, https://doi.org/10.1016/S0005-7967(96)00069-1. Alderson-Day, B., & Fernyhough, C. (2015). Inner speech: Development, cognitive functions, phenomenology, and neurobiology.Psychological Bulletin, 141(5), 931–965. https://doi.org/10.1037/bul0000021 Andrea N. Niles, Michelle G. Craske, Matthew D. Lieberman, Christopher Hur. (2015). Affect labeling enhances exposure effectiveness for public speaking anxiety, Behaviour Research and Therapy, Volume 68, Pages 27-36, ISSN 0005-7967, https://doi.org/10.1016/j.brat.2015.03.004. Antonis Hatzigeorgiadis, Nikos Zourbanos, Sofia Mpoumpaki, Yannis Theodorakis. (2009). Mechanisms underlying the self-talk–performance relationship: The effects of motivational self-talk on self-confidence and anxiety, Psychology of Sport and Exercise, Volume 10, Issue 1, Pages 186-192, ISSN 1469-0292, https://doi.org/10.1016/j.psychsport.2008.07.009. Arenella K, Steffen AM. Self-reassurance and self-efficacy for controlling upsetting thoughts predict depression, anxiety, and perceived stress in help-seeking female family caregivers. International Psychogeriatrics. 2020;32(2):229-240. https://doi.org/10.1017/S1041610219000565 Barcaccia, B., Hartstone, J.M., Pallini, S. et al. (2022). Mindfulness, Social Safeness and Self-Reassurance as Protective Factors and Self-Criticism and Revenge as Risk Factors for Depression and Anxiety Symptoms in Youth. Mindfulness 13, 674–684. https://doi.org/10.1007/s12671-021-01824-0 Barlow, D. H. (2002). Anxiety and its disorders: The nature and treatment of anxiety and panic (2nd ed.). The Guilford Press. https://psycnet.apa.org/record/2001-05896-000 Eysenck, Michael W.,Derakshan, Nazanin,Santos, Rita,Calvo, Manuel G. (2007. Anxiety and cognitive performance: Attentional control theory. Emotion, Vol 7(2), May 2007, 336-353 https://doi.org/10.1037/1528-3542.7.2.336 Kross, E., Bruehlman-Senecal, E., Park, J., Burson, A., Dougherty, A., Shablack, H., Bremner, R., Moser, J., & Ayduk, O. (2014). Self-talk as a regulatory mechanism: How you do it matters. Journal of Personality and Social Psychology, 106(2), 304–324. https://doi.org/10.1037/a0035173 Latinjak, A. T., Morin, A., Brinthaupt, T. M., Hardy, J., Hatzigeorgiadis, A., Kendall, P. C., Neck, C., Oliver, E. J., Puchalska-Wasyl, M. M., Tovares, A. V., & Winsler, A. (2023). Self-Talk: An Interdisciplinary Review and Transdisciplinary Model. Review of General Psychology, 27(4), 355-386. https://doi.org/10.1177/10892680231170263 (Original work published 2023) Lieberman, M. D., Eisenberger, N. I., Crockett, M. J., Tom, S. M., Pfeifer, J. H., & Way, B. M. (2007). Putting Feelings Into Words. Psychological Science, 18(5), 421-428. https://doi.org/10.1111/j.1467-9280.2007.01916.x Nicole Harte, Rachel V. Aaron, Kavya Bhattiprolu, Madelyne A. Bisby, Milena Gandy, Taylor Hathway, Blake F. Dear, Joanne Dudeney. (2024). The association between anxiety and depression symptoms and pain and function in adolescents and young adults with chronic pain: A meta-analysis, Journal of Psychosomatic Research, Volume 187, 111945, ISSN 0022-3999, https://doi.org/10.1016/j.jpsychores.2024.111945. Pan, X., Ding, W., Sun, X., Ji, C., Zhou, Q., Yan, C., … Luo, Y. (2021). Gray Matter Density of the Dorsomedial Prefrontal Cortex Mediates the Relationship Between Catastrophizing and Anxiety in Somatic Symptom Disorder. Neuropsychiatric Disease and Treatment, 17, 757–764. https://doi.org/10.2147/NDT.S296462 Petrini, Laura, and Lars Arendt-Nielsen. (2020). “Understanding Pain Catastrophizing: Putting Pieces Together.” Frontiers in Psychology, vol. 11, no. https://doi.org/10.3389/fpsyg.2020.603420. Rapee, R. M. (1993). The utilization of working memory by worry. Behaviour Research and Therapy, 31, 617– 620. https://doi: 10.1016/0005-7967(93)90114-a }} ==External links== * [https://www.cci.health.wa.gov.au/Resources/Looking-After-Yourself/Worry-and-Rumination Worry and Rumination] * [https://understandingvoices.com/exploring-voices/why-do-people-hear-voices/voices-and-inner-speech/ Voices and Inner Speech] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Anxiety]] [[Category:Motivation and emotion/Book/Inner voice]] 985xmll35he9y51zrmgh7nugsc5iq8k 0 323153 2834951 2834182 2026-09-29T01:59:32Z Jtneill 10242 /* Motivation */ [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|U3252175}} 2834951 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|U3280843}} # [[/Akrasia/]] - Why do people act against their better judgement? 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{{ME-By|P U3270518}} # [[/Psychedelic treatment of eating disorders/]] - How might psychedelic-assisted therapy influence psychological mechanisms involved in eating eating disorders? {{ME-By|Leilab23}} # [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|U3282586}} # [[/Romantic entertainment and love beliefs/]] - How does romantic entertainment influence beliefs and expectations about love and romantic relationships? {{ME-By|U3247927}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed?{{ME-By|U3279062}} # [[/Romantic rejection and emotional response/]] - What emotional responses does romantic rejection evoke and how can they be regulated? {{ME-By|U3275873}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|U3257744}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|Greg Philips}} # [[/Self-blame and emotion/]] - How does self-blame influence emotional responses to negative events? {{ME-By|U3281277}} # [[/Self-disclosure and emotional intimacy/]] - How does self-disclosure foster emotional closeness in relationships? {{ME-By|U3283302}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|Pinkk47}} # [[/Social connection and emotion regulation/]] - How do social relationships help regulate people's emotions? {{ME-By|U3284040}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|U3253354}} # [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|ChelsSchofield}} # [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|Safiaah}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|U3284437}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteers' subjective wellbeing? {{ME-By|U3216851}} ==Motivation and emotion== # [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|U3239431}} # [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|Mort006}} # [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|U3286962}} # [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|U3263365}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|U3306498}} # [[/Reinforcement sensitivity theory/]] - How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|UU3231448}} # [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|Aivy.uc.cbr}} # [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|Karabi Tasneem}} # [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|Med.011387}} [[Category:Motivation and emotion/Book/2026]] f3p2h6uvkyvkq7lnrqxop3yr75t10hv 2 325867 2834875 2830882 2026-09-28T19:19:54Z Dc.samizdat 2856930 /* Why the Equivalence Principle symmetry */ 2834875 wikitext text/x-wiki == Symmetry transformation group arising from the Laplace–Runge–Lenz vector == *{{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} <blockquote>Through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The ... infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector ... when combined with the rotation symmetries ... is shown to be the semi-direct product of <math>SO(3)</math> and <math>\mathbb{R^3}</math>. This stands in contrast to the <math>SO(4)</math> symmetry group generated by the LRL symmetries and the rotations.</blockquote> In other words, bodies in Kepler orbits are translating through 4-space via <math>QT</math>, breaking their stationary <math>Q^2</math> symmetry. The LRL direction vector is their <math>T</math>, their proper time vector. The Kepler problem in Newtonian mechanics has been widely studied and encompasses both a particle moving under inverse-square force and two bodies moving under mutual gravitational attraction. One of its most interesting features is the existence of the conserved Laplace–Runge–Lenz (LRL) vector (see e.g.  and references therein). It lies in the plane of motion, which is orthogonal to the angular momentum vector, and points in the direction given by the periapsis in the case of inverse-square motion and the position of closest separation in the case of two-body motion. This conserved vector has a long history, going back over three centuries, and plays a key role in understanding the shape and orientation of astronomical orbits in the solar system. Its quantum mechanical analog was important historically in the initial derivation of energy levels in the hydrogen atom (see e.g.  and references therein). In contrast to the conservation of energy and angular momentum in the Kepler problem, which come directly from Noether’s theorem using the manifest time-translation and rotation symmetries of Newtonian space, the LRL vector does not arise from any kinematic symmetry principle. Nevertheless, Noether’s theorem in reverse  states that every conserved quantity is generated by some infinitesimal symmetry transformation that leaves invariant the action principle for the Kepler problem. Since the LRL vector comprises three independent conserved quantities, its symmetry generator consists of three independent infinitesimal transformations, which were first worked out in Ref.. They turn out to involve the velocity variable, unlike the infinitesimal time-translation and rotation transformations which involve only the time and position variables. In particular, the LRL infinitesimal symmetry transformations are primary examples of what are called ''dynamical symmetries'', in contradistinction to ''point symmetries'' which are typified by infinitesimal time-translations and rotations. The only additional point symmetry admitted by the Kepler problem consists of a scaling of the time and position variables, but this scaling transformation does not leave the action principle invariant. The five point symmetries together with the three LRL symmetries in the Kepler problem form a 6-dimensional Lie subalgebra when the LRL symmetries are normalized to have the same physical dimension as the rotations. This subalgebra’s structure turns out to be <math>\mathfrak{so}(4)</math> when the energy is negative, and <math>\mathfrak{so}(3,1)</math> when the energy is positive . These Lie algebras are isomorphic to the symmetry algebra of isometries of a 3-sphere <math>S^3</math> in <math>\mathbb{R^4}</math> and a 3-hyperboloid <math>H^3</math> in Minkowski space <math>\mathbb{R^{3+1}}</math>. One explanation for the appearance of four-dimensional spaces is provided by Moser’s observation  that, in the negative energy case, the Kepler problem is equivalent to non-affine geodesic motion on <math>S^3</math> in a four-dimensional auxiliary space under stereographic projection. A kinematical description of the projection is given by identifying the stereographic coordinates as the components of the Kepler velocity variable. The rotations in the auxiliary space correspond in the Kepler problem to changing the angular momentum while keeping the energy unchanged. These rotations constitute a <math>SO(4)</math> transformation group that combines the ordinary physical <math>SO(3)</math> rotations and the LRL symmetry transformations. A similar equivalence holds in the positive energy case by considering non-affine geodesic motion on <math>H(3)</math>. One remaining question is how does the group generated by the LRL infinitesimal symmetry transformations act just on the time, position, and velocity variables in the Kepler problem. It is hard to find an explicit description of this group in the literature (although a description in terms of the coordinates in the auxiliary 4-dimensional space is known ). The first main result of the present work will be to derive the explicit form of the LRL symmetry transformation group in terms of the kinematic physical variables. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quantities is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated. Another remaining question, which is not commonly discussed in the literature, arises from the magnitude of LRL vector. This quantity is given by a scalar expression in terms of the energy and magnitude of the angular momentum. The corresponding infinitesimal symmetry transformation, coming from Noether’s theorem in reverse, is just a linear combination of the generators of time-translation and rotation in the plane of motion. Namely it is a point symmetry, involving only the time and position variables. On the one hand, this choice of normalization of the LRL vector is essential for the subalgebra generated by the LRL symmetries and rotations to be when energy is negative and when energy is positive. Yet on the other hand, it is a free choice, which raises the question of whether a different or perhaps simpler structure is possible for the resulting symmetry subalgebra if some other choice of normalization is made. A natural choice is to normalize it to be a unit vector, retaining only the physically relevant direction information contained in the LRL vector. This will be called the LRL direction vector. As a second main result, the components of the LRL direction vector will be shown to generate three infinitesimal symmetry transformations that commute with themselves. == Distribution of stars in our galaxy == The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity <math>c</math>. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion....{{Efn| When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits? That would be an expected consequence of the special rotational symmetry group of 4-space <math>SO(4)</math>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}} == Why the Equivalence Principle symmetry == How is the Weak Equivalence Principle (of Einstein) derived from Coxeter's group-theoretic formulation of the principle of relativity (his relation of congruent transformations in n-dimensional Euclidean space)? That is, what is the equivalence between rotations and translations, via their common basis in reflections? What exactly is the relative (i.e. symmetrical, interchangeable) relation between a translation and a rotation? == Light == === Polarization === How is polarized light oriented with respect to the proper time dimension? Is the magnetic field simply a fourth orthogonal orientation of the electric field? What is its orientation with respect to the proper time dimension? === Sails === On 3-sphere vs. 2-sphere - can you perhaps tack through 4-space on light pressure, if you can distinguish left and right hand radiation - relation to polarizations of light. Light pressure vs. particle pressure - <math>c\prime = 2c</math>. === Propagation kinematics through 4-space === 11-cell's role. Geodesics of truncated 5-cells. == 137 === * <math>137 = 5! + 4! - 3! - 2! + 1!</math> * a [[W:Pythagorean prime|Pythagorean prime]]: a prime number of the form <math>4n+1</math>, where <math>n=34</math>&nbsp;(<math>137=4\times 34+1</math>) or the sum of two squares <math>11^{2}+4^{2} = (121+16)</math>. * a combination of three terms <math>4^{3}+3^{4}-2^{3} = (64+81-8)</math>, cube of 4 + [[W:Triangular number|triangular number]] T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as free link) * The "73 calculator trick" involves picking any four-digit number, entering it twice to create an eight-digit number (e.g., 1234 becomes 12341234), then dividing by 137, and finally dividing that result by your original four-digit number to always get 73 as the answer. This works because repeating a four-digit decimal number N is the same as multiplying N by 10001, and 10001 is the product of 73 and 137 (73×137). == Galilean relativity in a space of four orthogonal dimensions == The paragraph ending in "That is all of physics" came to me on my mother's 101st birthday, just over a year after she died, like an example of rebirth in the spring. I say rebirth because it is probably not original, and even if it is, life must necessarily forget it someday, and just as necessarily rediscover it again someday afterwards. It seems to me to be as close an example to ''immortal life'' as I have encountered. That state, oxymoronic in nearly all senses I fear, may be said to exist where life discovers an immortal principle, and so (like the 11-cells) many lives distinct but not disjoint combine to give existence to an immortal object, where none could exist at all in singularity. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:44, 14 March 2024 (UTC) Writing this on the morning after my mother's 103rd birthday, after several related ideas finally assembled themselves in my mind in sleep last night, for the first time. This then is my first written observation of these several ideas that I have been pondering for years. The Higgs boson, the largest mediator of the fundamental force, has the mass/energy of the 120-cell. The photon as particle (as distinct from the free radiating photon as wave), the mediating boson of the electron, I think will turn out to have the mass/energy of the 11-cell. The 1/137 mass/energy relationship behind the ''e'' constant is due to the subsumption relationship between these two 4-polytopes and a third 4-polytope, the 4-simplex or 5-cell, which is the proton. Mass/energy is proportional to 4-content, and also a function of the isoclinic angle of rotation. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 03:57, 16 March 2026 (UTC) == Galaxies == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The universe is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own Milky Way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the Milky Way is not our whole galaxy; the majority of the other stars visible to the naked eye also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of 4-space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because it is a round sphere of nearby space surrounding us. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" where they appear to be more densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane, because we receive light signals from all directions in 4-space. We can see distant objects all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent 4-dimensional galaxy and out the other side). But we do not observe (or measure) the angles of objects above or below our hyperplane, because we conceptualize the volume of space around us as a 3-ball when it is actually a 4-ball. We unknowingly conflate all the objects we receive light from, projecting them into a single 3-ball, whether they are above, below or in our hyperplane. == The dimpled fabric of spacetime == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of a 3-sphere expanding radially at velocity <math>c</math>. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:15, 10 November 2024 (UTC) == John Stewart Bell by Adam Becker == A roulette wheel has three orthogonal dimensions of binary parity: high/low (a finite integer set partitioned in two), even/odd (a different partitioning of the integer set), and a third dimension red/black (which belong to a distinct sequence, and I dare say a distinct group, than the other two parities). "Polarization is a property of light -- light is an electromagnetic wave, and the polarization is the direction that the wave is doing its waving in. Polarization is sort of like a little arrow that each photon carries with it that can point in different dirdctions. But ... we can't actually tell what direction a photon's polarization arrow is pointing. All we can do is measure a photon's polarization along one particular axis at a time in a somewhat indirect [binary] manner ...." [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 21:36, 31 December 2024 (UTC) == Minkowski spacetime and Euclidean 4-space in relativity == ... Is there a configuration in Euclidean space where every observer, and every observed object, is moving at velocity <math>c</math>? Yes, there is one such configuration, in 4-dimensional Euclidean space. This configuration must be like the one described above, in our ''Abstract''. == The rate of atomic symmetry operations == ... == General relativity describes a curved 3-dimensional manifold embedded in Euclidean 4-dimensional space == ... == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a one or more concentric 4-polytopes of increasing radius. Each concentric shell is a single or compound [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating object, it is a kinematic one, because the nucleus of an actual atom of a distinct [[W:nucleon number|nucleon number]] contains a distinct number of orbiting 24-cell vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a compound of static 24-cells at any single instant in time, though their orbits do all the time. The physical configuration of a nucleus as concentric 24-cells can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituent protons and neutrons. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === Rotations === The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out. This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to properly visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of sensory experience in 4-dimensional space to draw upon. For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than a single rigid 24-cell. To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same shell without collisions? In adjacent shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[W:kinetics|kinetics]]. In four dimensions, we find that space is more commodious than we thought it was, and permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible more symmetric packings. Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, "frustrated" by their torsion. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]]....{{Efn|name=Petrie polygon of a honeycomb}} The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in cosmologic realm (at the surface boundaries of stars, and perhaps at the surface masses of hollow galaxies). But in the molecular realm we find only objects which occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a wholly imaginary one, though no less illuminating for the symmetry it reveals not having a realization as a 3-sphere of actual molecules. And might it not have a realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in flat 3-dimensional space. Suppose that the constituent particles of atoms actually close-pack the 3-sphere as helices, in the manner of Sadoc's imaginary twisted molecules, that is, as discrete Hopf fibrations of the atomic 4-polytope, where each discrete Hopf fibration corresponds to the distinct isoclinic rotation characteristic of a constituent particle. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:28, 8 November 2025 (UTC) === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Charge === Positive and negative charge are [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The perfection of Fuller's cyclic design|compression and tension]], respectively. ... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is an illustration of the [[5-cell]] labeling each of its <math>2^5</math> elements. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell & Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position.{{Efn|name=transformations}} .... === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... Coxeter orthogonal dimensional relativity furnishes the geometric spatial definition of the inertial-gravitational force. It does so by limiting free motion (in the sense of free-falling) to one rotation and one translation, making double rotations accelerations which involve an energy transition. === Physics === .... === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We are then led naturally to question whether the universe has only four dimensions. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From an ''n''-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2025|loc="A symmetrical arrangement of eleven 11-cells"}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Atomic elements === There are 120 5-cell protons in the 120-cell, and so there are 120 distinct atomic elements on the 3-sphere. === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== In the proper reference frame of some hypothetical observer in 4-dimensional Euclidean space, the Kepler problem has a solution in which all the planetary orbits are circles. This occurs because all bodies are always orbiting, each in some distinct isoclinic (equi-angled double) rotation. All atoms are rotating isoclinically at (double) the velocity <math>c</math>. In the observer's proper reference frame, the isoclinic double rotation of each of his own atoms is most conveniently represented as an equivalent screw displacement: a linear translation at velocity <math>c</math> (along the observer's proper time axis through 4-space), combined with a simple rotation at velocity <math>c</math> (of the atom internally). This representation, though proper, is subject to the over-simplification of special relativity, because it ignores the fact that the observer is himself moving on some circular orbit; he is not linearly translating through 4-space. When the observer's own orbit is also represented, the screw displacement becomes an elliptical double rotation, with a very slow, very large-radius simple circular orbit (almost a linear translation), combined with a very fast, very small-radius simple circular orbit of each atom internally. This is the observer's proper reference frame according to general relativity. An observer moving differently will perceive the motion of the observer and his atoms to be a different elliptical double rotation. We can find a proper reference frame, of some hypothetical observer moving hypothetically, in which the ratio of the radii of the two orthogonal rotations of the atom becomes any value we choose between 1 and <math>c</math>. In particular there exists, conceptually accessible to all observers independent of their motion, a distinct reference frame in which an atom is stationary, rotating isoclinically at (double) the velocity <math>c</math>. That is simply the stationary reference frame of the atom itself, such as the stationary reference frame of any observer and his own atoms. In 3-dimensional physics, it is known as the observer's proper inertial reference frame. But in 4-dimensional physics, that is ''not'' the most convenient or sensible reference frame in which to consider objects moving differently, in different proper reference frames. In 4-dimensional physics, a reference frame in which an observer and his atoms are translating linearly at velocity <math>c</math> is known as the observer's proper inertial reference frame. Any such proper reference frame makes a great deal more sense for the purpose of considering the motion of objects moving differently in 4-space, just as a heliocentric solar system makes a great deal more sense than a geocentric solar system for the purpose of considering the motion of the planets. ... ==Appendix== {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|last=Mac Gregor|first=Malcolm H.|title=The energy packet generator 137 and the key energy equation E_w + E_z = E_top|date=2017|publisher=El Mac Books|place=Santa Cruz, CA|author-link=W:Fine-structure constant|ref={{Mac Gregor 137|2017}}}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 41esefa86oatknd5zey7enombta8js4 0 326036 2834935 2805022 2026-09-29T00:12:27Z Doc James 28267 formatting 2834935 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Kenya (orthographic projection).svg|https://en.wikipedia.org/wiki/Kenya}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Kenya Kenya], officially the Republic of Kenya,is a country located in East Africa. Kenya is bordered by South Sudan to the northwest (though much of that border includes the disputed Ilemi Triangle), Ethiopia to the north, Somalia to the east, the Indian Ocean to the southeast, Tanzania to the southwest, and Lake Victoria and Uganda to the west. Kenya's ethnic groups typically speak their mother tongues within their own communities. The two official languages, English and Swahili, are used in varying degrees of fluency for communication with other populations. Kenya's constitution promotes and protects Kenyan Sign language. ==History== The Kenyan Ministry of Health launched a national plan for ear and hearing care in August of 2016. One of its mandates was to initiate and develop systems for early hearing loss detection and intervention. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Tucci|first2=Debara|last3=Lemons|first3=James|last4=Murila|first4=Florence|last5=Shepherd|first5=Susan|last6=Mwangi|first6=Moses|last7=Macharia|first7=Isaac|last8=Ayugi|first8=John|date=2024-03|title=Newborn and infant hearing screening for early detection of hearing loss in Nairobi, Kenya|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11217834/|journal=African Health Sciences|volume=24|issue=1|pages=228–238|doi=10.4314/ahs.v24i1.28|issn=1729-0503|pmc=11217834|pmid=38962342}}</ref> ==Frequency== The prevalence of moderate or severe hearing loss in Kenya is estimated to be 14 in every 1,000 children.<ref>{{Cite journal|last=Mung'ala-Odera|first=V|last2=Meehan|first2=R|last3=Njuguna|first3=P|last4=Mturi|first4=N|last5=Alcock|first5=Kj|last6=Newton|first6=Crjc|date=2006-06-01|title=Prevalence and risk factors of neurological disability and impairment in children living in rural Kenya|url=http://academic.oup.com/ije/article/35/3/683/735669/Prevalence-and-risk-factors-of-neurological|journal=International Journal of Epidemiology|language=en|volume=35|issue=3|pages=683–688|doi=10.1093/ije/dyl023|issn=1464-3685}}</ref> Risk factors for acquired hearing loss include congenital infections (e.g., toxoplasmosis, rubella, cytomegalovirus, and herpes), premature birth, in utero exposure to HIV, neonatal jaundice, recurrent ear infections, HIV infection, bacterial meningitis, and ototoxic medications such as those used to treat tuberculosis, pneumonia, or malaria. <ref>{{Cite journal|last=Paradowska|first=Edyta|last2=Jabłońska|first2=Agnieszka|last3=Studzińska|first3=Mirosława|last4=Kasztelewicz|first4=Beata|last5=Zawilińska|first5=Barbara|last6=Wiśniewska‐Ligier|first6=Małgorzata|last7=Dzierżanowska‐Fangrat|first7=Katarzyna|last8=Woźniakowska‐Gęsicka|first8=Teresa|last9=Kosz‐Vnenchak|first9=Magdalena|date=2014-08|title=Cytomegalovirus glycoprotein H genotype distribution and the relationship with hearing loss in children|url=https://onlinelibrary.wiley.com/doi/10.1002/jmv.23906|journal=Journal of Medical Virology|language=en|volume=86|issue=8|pages=1421–1427|doi=10.1002/jmv.23906|issn=0146-6615}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> The prevalence of disabling hearing loss in Sub-Saharan Africa is estimated to be 4.55% for all ages. The World Health Organization (WHO) classifies hearing loss as Mild (26-40 dBHL), Moderate (41-60 dBHL), Severe (61-80dBHL) and Profound (≥81dBHL). Hearing loss is a hidden disability as it is invisible and therefore often goes undetected and untreated. The prevalence of permanent hearing loss in developing nations is between 3-6/1000 in newborn babies while in developed countries the range is between 1-3/1000. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> {{HTitle|Information About Audiology}} There is limited information available specifically about the practice of audiology in Kenya. A 2018 study conducted in Malindi, Kenya, demonstrated the feasibility of using non-otolaryngology-based hearing screening programs. Community health workers and nursing staff successfully performed audiograms and otoscopic endoscopy using mobile technology, indicating potential for expanding audiology services through community-based initiatives.<ref>{{Cite journal|last=Jayawardena|first=Asitha D. L.|last2=Kahue|first2=Charissa N.|last3=Cummins|first3=Samantha M.|last4=Netterville|first4=James L.|date=2018-01|title=Expanding the Capacity of Otolaryngologists in Kenya through Mobile Technology|url=https://aao-hnsfjournals.onlinelibrary.wiley.com/doi/10.1177/2473974X18766824|journal=OTO Open|language=en|volume=2|issue=1|doi=10.1177/2473974X18766824|issn=2473-974X}}</ref> In 2024 a joint Nairobi-and Seattle-based group examined and published strategies for developing national EHC programs in Kenya. Recommendations include leveraging existing health policies, strengthening referral pathways, training healthcare workers, and engaging local partners for phased scale-up. These strategies aim to improve access to screening, diagnostics, management, and developmental support for hearing loss.<ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> {{HTitle|Scope of Practice and Licensing}} As of 2025, there is one training program for clinical audiologists and speech-language pathologists through the University of Nairobi. The degree is a Bachelor is Science in Audiology, Speech, and Language Pathology (B.ASLP). The program is 4 years and the final year includes a clinical internship. <ref>https://surgery.uonbi.ac.ke/programs/bachelor-science-audiology-speech-and-language-pathology-baslp</ref> {{HTitle|Professional and Regulatory Bodies}} *The Kenya Ear Nose and Throat Society (KENTS) founded in 1988, represents otorhinolaryngologists, ENT Clinical Officers, Audiologists, and ENT nursing officers. It provides continuing medical education and professional development for its members. *The Association of Speech and Language Therapists Kenya works with audiologists on hearing care initiatives and is part of the national Ear and Hearing Care strategic planning team. * The Kenya Medical Practitioners and Dentists Board (KMPDB) is the statutory body that regulates audiology centres and hearing aid practices in Kenya. It board regulates the training and practice of medicine, dentistry, and community oral health, and audiology. Audiology centers must be registered with them to operate legally. {{HTitle|Ongoing audiology research}} Research in Kenya is focusing on addressing hearing care through early detection, community involvement, policy development, and innovative technologies and treatment approaches. Continued efforts are needed to overcome existing barriers and ensure comprehensive, accessible hearing care for all. {{HTitle|Challenges, Opportunities and Notes}} Challenges such as lack of equipment, training, and follow-up systems are significant barriers to effective hearing care. The majority of audiology services in Kenya are centralized in the capital, Nairobi. Families living outside of the capital are generally underserved. Additionally, there is limited access to services due to a small number of providers. These barriers lead to low awareness, delayed diagnosis and gaps in hearing care in Kenya. <ref> https://childrenwithhearingloss.org/kenya-program/</ref> The improvement of data management and follow-up systems, and leveraging local partnerships for a phased scale-up are opportunities that could significantly improve the provision of hearing services. {{HTitle|Audiology Charities}} * Kenya Society for Deaf Children (KSDC) * Deaf Empowerment Society of Kenya (DESK) * Kentalis International Foundation] * Deaf Child Worldwide] (international organization working in Kenya) {{HTitle|References}} {{reflist}} === '''External links'''=== '' Professional associations'' * Kenya Ear Nose and Throat Society ([https://kenyaentsociety.or.ke/ KENTS]) * Association of Speech and Language Therapists Kenya ([https://asltk.blogspot.com/ ASLTK]) * Kenya Medical Practitioners and Dentists Board ([https://kmpdc.go.ke/ KMPDB]) '' Charities'' * Kenya Society for Deaf Children ([https://deafchildrensociety-kenya.org/ KSDC]) * Deaf Empowerment Society of Kenya ([https://deskkenya.org/ DESK][https://www.kentalis.com/]) * [https://www.kentalis.com/ Kentalis International Foundation] * Deaf Child Worldwide (international organization working in Kenya, [https://www.ndcs.org.uk/deaf-child-worldwide/ NDCS]) *[https://www.kentalis.com/] {{:Global Audiology/Authors-1|Gabriella Green|https://www.linkedin.com/in/gabriellagreen206/}} [[Category:Audiology]] [[Category:Kenya]] gqa61pu2phci85woqqgv8kh4xid8b9e 2834936 2834935 2026-09-29T00:15:41Z Doc James 28267 2834936 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Kenya (orthographic projection).svg|https://en.wikipedia.org/wiki/Kenya}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Kenya Kenya], officially the Republic of Kenya,is a country located in East Africa. Kenya is bordered by South Sudan to the northwest (though much of that border includes the disputed Ilemi Triangle), Ethiopia to the north, Somalia to the east, the Indian Ocean to the southeast, Tanzania to the southwest, and Lake Victoria and Uganda to the west. Kenya's ethnic groups typically speak their mother tongues within their own communities. The two official languages, English and Swahili, are used in varying degrees of fluency for communication with other populations. Kenya's constitution promotes and protects Kenyan Sign language. ==History== The Kenyan Ministry of Health launched a national plan for ear and hearing care in August of 2016. One of its mandates was to initiate and develop systems for early hearing loss detection and intervention. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Tucci|first2=Debara|last3=Lemons|first3=James|last4=Murila|first4=Florence|last5=Shepherd|first5=Susan|last6=Mwangi|first6=Moses|last7=Macharia|first7=Isaac|last8=Ayugi|first8=John|date=2024-03|title=Newborn and infant hearing screening for early detection of hearing loss in Nairobi, Kenya|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11217834/|journal=African Health Sciences|volume=24|issue=1|pages=228–238|doi=10.4314/ahs.v24i1.28|issn=1729-0503|pmc=11217834|pmid=38962342}}</ref> ==Frequency== The prevalence of moderate or severe hearing loss in Kenya is estimated to be 14 in every 1,000 children.<ref>{{Cite journal|last=Mung'ala-Odera|first=V|last2=Meehan|first2=R|last3=Njuguna|first3=P|last4=Mturi|first4=N|last5=Alcock|first5=Kj|last6=Newton|first6=Crjc|date=2006-06-01|title=Prevalence and risk factors of neurological disability and impairment in children living in rural Kenya|url=http://academic.oup.com/ije/article/35/3/683/735669/Prevalence-and-risk-factors-of-neurological|journal=International Journal of Epidemiology|language=en|volume=35|issue=3|pages=683–688|doi=10.1093/ije/dyl023|issn=1464-3685}}</ref> Risk factors for acquired hearing loss include congenital infections (e.g., toxoplasmosis, rubella, cytomegalovirus, and herpes), premature birth, in utero exposure to HIV, neonatal jaundice, recurrent ear infections, HIV infection, bacterial meningitis, and ototoxic medications such as those used to treat tuberculosis, pneumonia, or malaria. <ref>{{Cite journal|last=Paradowska|first=Edyta|last2=Jabłońska|first2=Agnieszka|last3=Studzińska|first3=Mirosława|last4=Kasztelewicz|first4=Beata|last5=Zawilińska|first5=Barbara|last6=Wiśniewska‐Ligier|first6=Małgorzata|last7=Dzierżanowska‐Fangrat|first7=Katarzyna|last8=Woźniakowska‐Gęsicka|first8=Teresa|last9=Kosz‐Vnenchak|first9=Magdalena|date=2014-08|title=Cytomegalovirus glycoprotein H genotype distribution and the relationship with hearing loss in children|url=https://onlinelibrary.wiley.com/doi/10.1002/jmv.23906|journal=Journal of Medical Virology|language=en|volume=86|issue=8|pages=1421–1427|doi=10.1002/jmv.23906|issn=0146-6615}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> The prevalence of disabling hearing loss in Sub-Saharan Africa is estimated to be 4.55% for all ages. The World Health Organization (WHO) classifies hearing loss as Mild (26-40 dBHL), Moderate (41-60 dBHL), Severe (61-80dBHL) and Profound (≥81dBHL). Hearing loss is a hidden disability as it is invisible and therefore often goes undetected and untreated. The prevalence of permanent hearing loss in developing nations is between 3-6/1000 in newborn babies while in developed countries the range is between 1-3/1000. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> ==About Audiology== There is limited information available specifically about the practice of audiology in Kenya. A 2018 study conducted in Malindi, Kenya, demonstrated the feasibility of using non-otolaryngology-based hearing screening programs. Community health workers and nursing staff successfully performed audiograms and otoscopic endoscopy using mobile technology, indicating potential for expanding audiology services through community-based initiatives.<ref>{{Cite journal|last=Jayawardena|first=Asitha D. L.|last2=Kahue|first2=Charissa N.|last3=Cummins|first3=Samantha M.|last4=Netterville|first4=James L.|date=2018-01|title=Expanding the Capacity of Otolaryngologists in Kenya through Mobile Technology|url=https://aao-hnsfjournals.onlinelibrary.wiley.com/doi/10.1177/2473974X18766824|journal=OTO Open|language=en|volume=2|issue=1|doi=10.1177/2473974X18766824|issn=2473-974X}}</ref> In 2024 a joint Nairobi-and Seattle-based group examined and published strategies for developing national EHC programs in Kenya. Recommendations include leveraging existing health policies, strengthening referral pathways, training healthcare workers, and engaging local partners for phased scale-up. These strategies aim to improve access to screening, diagnostics, management, and developmental support for hearing loss.<ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> ==Scope of practice and licensing== As of 2025, there is one training program for clinical audiologists and speech-language pathologists through the University of Nairobi. The degree is a Bachelor is Science in Audiology, Speech, and Language Pathology (B.ASLP). The program is 4 years and the final year includes a clinical internship. <ref>https://surgery.uonbi.ac.ke/programs/bachelor-science-audiology-speech-and-language-pathology-baslp</ref> ==Professional and Regulatory Bodies== *The Kenya Ear Nose and Throat Society (KENTS) founded in 1988, represents otorhinolaryngologists, ENT Clinical Officers, Audiologists, and ENT nursing officers. It provides continuing medical education and professional development for its members. *The Association of Speech and Language Therapists Kenya works with audiologists on hearing care initiatives and is part of the national Ear and Hearing Care strategic planning team. * The Kenya Medical Practitioners and Dentists Board (KMPDB) is the statutory body that regulates audiology centres and hearing aid practices in Kenya. It board regulates the training and practice of medicine, dentistry, and community oral health, and audiology. Audiology centers must be registered with them to operate legally. ==Ongoing research== Research in Kenya is focusing on addressing hearing care through early detection, community involvement, policy development, and innovative technologies and treatment approaches. Continued efforts are needed to overcome existing barriers and ensure comprehensive, accessible hearing care for all. ==Challenges== Challenges such as lack of equipment, training, and follow-up systems are significant barriers to effective hearing care. The majority of audiology services in Kenya are centralized in the capital, Nairobi. Families living outside of the capital are generally underserved. Additionally, there is limited access to services due to a small number of providers. These barriers lead to low awareness, delayed diagnosis and gaps in hearing care in Kenya. <ref> https://childrenwithhearingloss.org/kenya-program/</ref> The improvement of data management and follow-up systems, and leveraging local partnerships for a phased scale-up are opportunities that could significantly improve the provision of hearing services. ==Charities== * Kenya Society for Deaf Children (KSDC) * Deaf Empowerment Society of Kenya (DESK) * Kentalis International Foundation] * Deaf Child Worldwide] (international organization working in Kenya) ==References== {{reflist}} ==External links== '' Professional associations'' * Kenya Ear Nose and Throat Society ([https://kenyaentsociety.or.ke/ KENTS]) * Association of Speech and Language Therapists Kenya ([https://asltk.blogspot.com/ ASLTK]) * Kenya Medical Practitioners and Dentists Board ([https://kmpdc.go.ke/ KMPDB]) '' Charities'' * Kenya Society for Deaf Children ([https://deafchildrensociety-kenya.org/ KSDC]) * Deaf Empowerment Society of Kenya ([https://deskkenya.org/ DESK][https://www.kentalis.com/]) * [https://www.kentalis.com/ Kentalis International Foundation] * Deaf Child Worldwide (international organization working in Kenya, [https://www.ndcs.org.uk/deaf-child-worldwide/ NDCS]) *[https://www.kentalis.com/ Kentalis] {{:Global Audiology/Authors-1|Gabriella Green|https://www.linkedin.com/in/gabriellagreen206/}} [[Category:Audiology]] [[Category:Kenya]] mv2t68gw4sba5apdignfa9lg2x13yh7 2834937 2834936 2026-09-29T00:16:36Z Doc James 28267 2834937 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Kenya (orthographic projection).svg|https://en.wikipedia.org/wiki/Kenya}} ==General information== [https://en.wikipedia.org/wiki/Kenya Kenya], officially the Republic of Kenya, is a country located in East Africa. Kenya is bordered by South Sudan to the northwest (though much of that border includes the disputed Ilemi Triangle), Ethiopia to the north, Somalia to the east, the Indian Ocean to the southeast, Tanzania to the southwest, and Lake Victoria and Uganda to the west. Kenya's ethnic groups typically speak their mother tongues within their own communities. The two official languages, English and Swahili, are used in varying degrees of fluency for communication with other populations. Kenya's constitution promotes and protects Kenyan Sign language. ==History== The Kenyan Ministry of Health launched a national plan for ear and hearing care in August of 2016. One of its mandates was to initiate and develop systems for early hearing loss detection and intervention. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Tucci|first2=Debara|last3=Lemons|first3=James|last4=Murila|first4=Florence|last5=Shepherd|first5=Susan|last6=Mwangi|first6=Moses|last7=Macharia|first7=Isaac|last8=Ayugi|first8=John|date=2024-03|title=Newborn and infant hearing screening for early detection of hearing loss in Nairobi, Kenya|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11217834/|journal=African Health Sciences|volume=24|issue=1|pages=228–238|doi=10.4314/ahs.v24i1.28|issn=1729-0503|pmc=11217834|pmid=38962342}}</ref> ==Frequency== The prevalence of moderate or severe hearing loss in Kenya is estimated to be 14 in every 1,000 children.<ref>{{Cite journal|last=Mung'ala-Odera|first=V|last2=Meehan|first2=R|last3=Njuguna|first3=P|last4=Mturi|first4=N|last5=Alcock|first5=Kj|last6=Newton|first6=Crjc|date=2006-06-01|title=Prevalence and risk factors of neurological disability and impairment in children living in rural Kenya|url=http://academic.oup.com/ije/article/35/3/683/735669/Prevalence-and-risk-factors-of-neurological|journal=International Journal of Epidemiology|language=en|volume=35|issue=3|pages=683–688|doi=10.1093/ije/dyl023|issn=1464-3685}}</ref> Risk factors for acquired hearing loss include congenital infections (e.g., toxoplasmosis, rubella, cytomegalovirus, and herpes), premature birth, in utero exposure to HIV, neonatal jaundice, recurrent ear infections, HIV infection, bacterial meningitis, and ototoxic medications such as those used to treat tuberculosis, pneumonia, or malaria. <ref>{{Cite journal|last=Paradowska|first=Edyta|last2=Jabłońska|first2=Agnieszka|last3=Studzińska|first3=Mirosława|last4=Kasztelewicz|first4=Beata|last5=Zawilińska|first5=Barbara|last6=Wiśniewska‐Ligier|first6=Małgorzata|last7=Dzierżanowska‐Fangrat|first7=Katarzyna|last8=Woźniakowska‐Gęsicka|first8=Teresa|last9=Kosz‐Vnenchak|first9=Magdalena|date=2014-08|title=Cytomegalovirus glycoprotein H genotype distribution and the relationship with hearing loss in children|url=https://onlinelibrary.wiley.com/doi/10.1002/jmv.23906|journal=Journal of Medical Virology|language=en|volume=86|issue=8|pages=1421–1427|doi=10.1002/jmv.23906|issn=0146-6615}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> The prevalence of disabling hearing loss in Sub-Saharan Africa is estimated to be 4.55% for all ages. The World Health Organization (WHO) classifies hearing loss as Mild (26-40 dBHL), Moderate (41-60 dBHL), Severe (61-80dBHL) and Profound (≥81dBHL). Hearing loss is a hidden disability as it is invisible and therefore often goes undetected and untreated. The prevalence of permanent hearing loss in developing nations is between 3-6/1000 in newborn babies while in developed countries the range is between 1-3/1000. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> ==About Audiology== There is limited information available specifically about the practice of audiology in Kenya. A 2018 study conducted in Malindi, Kenya, demonstrated the feasibility of using non-otolaryngology-based hearing screening programs. Community health workers and nursing staff successfully performed audiograms and otoscopic endoscopy using mobile technology, indicating potential for expanding audiology services through community-based initiatives.<ref>{{Cite journal|last=Jayawardena|first=Asitha D. L.|last2=Kahue|first2=Charissa N.|last3=Cummins|first3=Samantha M.|last4=Netterville|first4=James L.|date=2018-01|title=Expanding the Capacity of Otolaryngologists in Kenya through Mobile Technology|url=https://aao-hnsfjournals.onlinelibrary.wiley.com/doi/10.1177/2473974X18766824|journal=OTO Open|language=en|volume=2|issue=1|doi=10.1177/2473974X18766824|issn=2473-974X}}</ref> In 2024 a joint Nairobi-and Seattle-based group examined and published strategies for developing national EHC programs in Kenya. Recommendations include leveraging existing health policies, strengthening referral pathways, training healthcare workers, and engaging local partners for phased scale-up. These strategies aim to improve access to screening, diagnostics, management, and developmental support for hearing loss.<ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> ==Scope of practice and licensing== As of 2025, there is one training program for clinical audiologists and speech-language pathologists through the University of Nairobi. The degree is a Bachelor is Science in Audiology, Speech, and Language Pathology (B.ASLP). The program is 4 years and the final year includes a clinical internship. <ref>https://surgery.uonbi.ac.ke/programs/bachelor-science-audiology-speech-and-language-pathology-baslp</ref> ==Professional and Regulatory Bodies== *The Kenya Ear Nose and Throat Society (KENTS) founded in 1988, represents otorhinolaryngologists, ENT Clinical Officers, Audiologists, and ENT nursing officers. It provides continuing medical education and professional development for its members. *The Association of Speech and Language Therapists Kenya works with audiologists on hearing care initiatives and is part of the national Ear and Hearing Care strategic planning team. * The Kenya Medical Practitioners and Dentists Board (KMPDB) is the statutory body that regulates audiology centres and hearing aid practices in Kenya. It board regulates the training and practice of medicine, dentistry, and community oral health, and audiology. Audiology centers must be registered with them to operate legally. ==Ongoing research== Research in Kenya is focusing on addressing hearing care through early detection, community involvement, policy development, and innovative technologies and treatment approaches. Continued efforts are needed to overcome existing barriers and ensure comprehensive, accessible hearing care for all. ==Challenges== Challenges such as lack of equipment, training, and follow-up systems are significant barriers to effective hearing care. The majority of audiology services in Kenya are centralized in the capital, Nairobi. Families living outside of the capital are generally underserved. Additionally, there is limited access to services due to a small number of providers. These barriers lead to low awareness, delayed diagnosis and gaps in hearing care in Kenya. <ref> https://childrenwithhearingloss.org/kenya-program/</ref> The improvement of data management and follow-up systems, and leveraging local partnerships for a phased scale-up are opportunities that could significantly improve the provision of hearing services. ==Charities== * Kenya Society for Deaf Children (KSDC) * Deaf Empowerment Society of Kenya (DESK) * Kentalis International Foundation] * Deaf Child Worldwide] (international organization working in Kenya) ==References== {{reflist}} ==External links== '' Professional associations'' * Kenya Ear Nose and Throat Society ([https://kenyaentsociety.or.ke/ KENTS]) * Association of Speech and Language Therapists Kenya ([https://asltk.blogspot.com/ ASLTK]) * Kenya Medical Practitioners and Dentists Board ([https://kmpdc.go.ke/ KMPDB]) '' Charities'' * Kenya Society for Deaf Children ([https://deafchildrensociety-kenya.org/ KSDC]) * Deaf Empowerment Society of Kenya ([https://deskkenya.org/ DESK][https://www.kentalis.com/]) * [https://www.kentalis.com/ Kentalis International Foundation] * Deaf Child Worldwide (international organization working in Kenya, [https://www.ndcs.org.uk/deaf-child-worldwide/ NDCS]) *[https://www.kentalis.com/ Kentalis] {{:Global Audiology/Authors-1|Gabriella Green|https://www.linkedin.com/in/gabriellagreen206/}} [[Category:Audiology]] [[Category:Kenya]] iizqym8bv72rf9ro733hwnoaeqzr1mr 2834938 2834937 2026-09-29T00:18:17Z Doc James 28267 2834938 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Kenya (orthographic projection).svg|https://en.wikipedia.org/wiki/Kenya}} ==General information== [https://en.wikipedia.org/wiki/Kenya Kenya], officially the Republic of Kenya, is a country located in East Africa. Kenya is bordered by South Sudan to the northwest (though much of that border includes the disputed Ilemi Triangle), Ethiopia to the north, Somalia to the east, the Indian Ocean to the southeast, Tanzania to the southwest, and Lake Victoria and Uganda to the west. Kenya's ethnic groups typically speak their mother tongues within their own communities. The two official languages, English and Swahili, are used in varying degrees of fluency for communication with other populations. Kenya's constitution promotes and protects Kenyan Sign language. ==History== The Kenyan Ministry of Health launched a national plan for ear and hearing care in August of 2016. One of its mandates was to initiate and develop systems for early hearing loss detection and intervention. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Tucci|first2=Debara|last3=Lemons|first3=James|last4=Murila|first4=Florence|last5=Shepherd|first5=Susan|last6=Mwangi|first6=Moses|last7=Macharia|first7=Isaac|last8=Ayugi|first8=John|date=2024-03|title=Newborn and infant hearing screening for early detection of hearing loss in Nairobi, Kenya|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11217834/|journal=African Health Sciences|volume=24|issue=1|pages=228–238|doi=10.4314/ahs.v24i1.28|issn=1729-0503|pmc=11217834|pmid=38962342}}</ref> ==Frequency== The prevalence of moderate or severe hearing loss in Kenya is estimated to be 14 in every 1,000 children.<ref>{{Cite journal|last=Mung'ala-Odera|first=V|last2=Meehan|first2=R|last3=Njuguna|first3=P|last4=Mturi|first4=N|last5=Alcock|first5=Kj|last6=Newton|first6=Crjc|date=2006-06-01|title=Prevalence and risk factors of neurological disability and impairment in children living in rural Kenya|url=http://academic.oup.com/ije/article/35/3/683/735669/Prevalence-and-risk-factors-of-neurological|journal=International Journal of Epidemiology|language=en|volume=35|issue=3|pages=683–688|doi=10.1093/ije/dyl023|issn=1464-3685}}</ref> Risk factors for acquired hearing loss include congenital infections (e.g., toxoplasmosis, rubella, cytomegalovirus, and herpes), premature birth, in utero exposure to HIV, neonatal jaundice, recurrent ear infections, HIV infection, bacterial meningitis, and ototoxic medications such as those used to treat tuberculosis, pneumonia, or malaria. <ref>{{Cite journal|last=Paradowska|first=Edyta|last2=Jabłońska|first2=Agnieszka|last3=Studzińska|first3=Mirosława|last4=Kasztelewicz|first4=Beata|last5=Zawilińska|first5=Barbara|last6=Wiśniewska‐Ligier|first6=Małgorzata|last7=Dzierżanowska‐Fangrat|first7=Katarzyna|last8=Woźniakowska‐Gęsicka|first8=Teresa|last9=Kosz‐Vnenchak|first9=Magdalena|date=2014-08|title=Cytomegalovirus glycoprotein H genotype distribution and the relationship with hearing loss in children|url=https://onlinelibrary.wiley.com/doi/10.1002/jmv.23906|journal=Journal of Medical Virology|language=en|volume=86|issue=8|pages=1421–1427|doi=10.1002/jmv.23906|issn=0146-6615}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> The prevalence of disabling hearing loss in Sub-Saharan Africa is estimated to be 4.55% for all ages. The World Health Organization (WHO) classifies hearing loss as Mild (26-40 dBHL), Moderate (41-60 dBHL), Severe (61-80dBHL) and Profound (≥81dBHL). Hearing loss is a hidden disability as it is invisible and therefore often goes undetected and untreated. The prevalence of permanent hearing loss in developing nations is between 3-6/1000 in newborn babies while in developed countries the range is between 1-3/1000. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> ==About Audiology== There is limited information available specifically about the practice of audiology in Kenya. A 2018 study conducted in Malindi, Kenya, demonstrated the feasibility of using non-otolaryngology-based hearing screening programs. Community health workers and nursing staff successfully performed audiograms and otoscopic endoscopy using mobile technology, indicating potential for expanding audiology services through community-based initiatives.<ref>{{Cite journal|last=Jayawardena|first=Asitha D. L.|last2=Kahue|first2=Charissa N.|last3=Cummins|first3=Samantha M.|last4=Netterville|first4=James L.|date=2018-01|title=Expanding the Capacity of Otolaryngologists in Kenya through Mobile Technology|url=https://aao-hnsfjournals.onlinelibrary.wiley.com/doi/10.1177/2473974X18766824|journal=OTO Open|language=en|volume=2|issue=1|doi=10.1177/2473974X18766824|issn=2473-974X}}</ref> In 2024 a joint Nairobi-and Seattle-based group examined and published strategies for developing national EHC programs in Kenya. Recommendations include leveraging existing health policies, strengthening referral pathways, training healthcare workers, and engaging local partners for phased scale-up. These strategies aim to improve access to screening, diagnostics, management, and developmental support for hearing loss.<ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> ==Scope of practice and licensing== As of 2025, there is one training program for clinical audiologists and speech-language pathologists through the University of Nairobi. The degree is a Bachelor is Science in Audiology, Speech, and Language Pathology (B.ASLP). The program is 4 years and the final year includes a clinical internship. <ref>https://surgery.uonbi.ac.ke/programs/bachelor-science-audiology-speech-and-language-pathology-baslp</ref> ===Professional and regulatory bodies=== *The Kenya Ear Nose and Throat Society (KENTS) founded in 1988, represents otorhinolaryngologists, ENT Clinical Officers, Audiologists, and ENT nursing officers. It provides continuing medical education and professional development for its members. *The Association of Speech and Language Therapists Kenya works with audiologists on hearing care initiatives and is part of the national Ear and Hearing Care strategic planning team. * The Kenya Medical Practitioners and Dentists Board (KMPDB) is the statutory body that regulates audiology centres and hearing aid practices in Kenya. It board regulates the training and practice of medicine, dentistry, and community oral health, and audiology. Audiology centers must be registered with them to operate legally. ==Ongoing research== Research in Kenya is focusing on addressing hearing care through early detection, community involvement, policy development, and innovative technologies and treatment approaches. Continued efforts are needed to overcome existing barriers and ensure comprehensive, accessible hearing care for all. ==Challenges== Challenges such as lack of equipment, training, and follow-up systems are significant barriers to effective hearing care. The majority of audiology services in Kenya are centralized in the capital, Nairobi. Families living outside of the capital are generally underserved. Additionally, there is limited access to services due to a small number of providers. These barriers lead to low awareness, delayed diagnosis and gaps in hearing care in Kenya. <ref> https://childrenwithhearingloss.org/kenya-program/</ref> The improvement of data management and follow-up systems, and leveraging local partnerships for a phased scale-up are opportunities that could significantly improve the provision of hearing services. ==Charities== * Kenya Society for Deaf Children (KSDC) * Deaf Empowerment Society of Kenya (DESK) * Kentalis International Foundation] * Deaf Child Worldwide] (international organization working in Kenya) ==References== {{reflist}} ==External links== '' Professional associations'' * Kenya Ear Nose and Throat Society ([https://kenyaentsociety.or.ke/ KENTS]) * Association of Speech and Language Therapists Kenya ([https://asltk.blogspot.com/ ASLTK]) * Kenya Medical Practitioners and Dentists Board ([https://kmpdc.go.ke/ KMPDB]) '' Charities'' * Kenya Society for Deaf Children ([https://deafchildrensociety-kenya.org/ KSDC]) * Deaf Empowerment Society of Kenya ([https://deskkenya.org/ DESK][https://www.kentalis.com/]) * [https://www.kentalis.com/ Kentalis International Foundation] * Deaf Child Worldwide (international organization working in Kenya, [https://www.ndcs.org.uk/deaf-child-worldwide/ NDCS]) *[https://www.kentalis.com/ Kentalis] {{:Global Audiology/Authors-1|Gabriella Green|https://www.linkedin.com/in/gabriellagreen206/}} [[Category:Audiology]] [[Category:Kenya]] b081dbsj5hhlfojoyg2zbv1qj834vgp 2834954 2834938 2026-09-29T02:42:33Z Doc James 28267 already present, reverted self 2834954 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Kenya (orthographic projection).svg|https://en.wikipedia.org/wiki/Kenya}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Kenya Kenya], officially the Republic of Kenya,is a country located in East Africa. Kenya is bordered by South Sudan to the northwest (though much of that border includes the disputed Ilemi Triangle), Ethiopia to the north, Somalia to the east, the Indian Ocean to the southeast, Tanzania to the southwest, and Lake Victoria and Uganda to the west. Kenya's ethnic groups typically speak their mother tongues within their own communities. The two official languages, English and Swahili, are used in varying degrees of fluency for communication with other populations. Kenya's constitution promotes and protects Kenyan Sign language. {{HTitle|History of Audiology}} The Kenyan Ministry of Health launched a national plan for ear and hearing care in August of 2016. One of its mandates was to initiate and develop systems for early hearing loss detection and intervention. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Tucci|first2=Debara|last3=Lemons|first3=James|last4=Murila|first4=Florence|last5=Shepherd|first5=Susan|last6=Mwangi|first6=Moses|last7=Macharia|first7=Isaac|last8=Ayugi|first8=John|date=2024-03|title=Newborn and infant hearing screening for early detection of hearing loss in Nairobi, Kenya|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC11217834/|journal=African Health Sciences|volume=24|issue=1|pages=228–238|doi=10.4314/ahs.v24i1.28|issn=1729-0503|pmc=11217834|pmid=38962342}}</ref> {{HTitle|Incidence and Prevalence of Hearing Loss}} The prevalence of moderate or severe hearing loss in Kenya is estimated to be 14 in every 1,000 children.<ref>{{Cite journal|last=Mung'ala-Odera|first=V|last2=Meehan|first2=R|last3=Njuguna|first3=P|last4=Mturi|first4=N|last5=Alcock|first5=Kj|last6=Newton|first6=Crjc|date=2006-06-01|title=Prevalence and risk factors of neurological disability and impairment in children living in rural Kenya|url=http://academic.oup.com/ije/article/35/3/683/735669/Prevalence-and-risk-factors-of-neurological|journal=International Journal of Epidemiology|language=en|volume=35|issue=3|pages=683–688|doi=10.1093/ije/dyl023|issn=1464-3685}}</ref> Risk factors for acquired hearing loss include congenital infections (e.g., toxoplasmosis, rubella, cytomegalovirus, and herpes), premature birth, in utero exposure to HIV, neonatal jaundice, recurrent ear infections, HIV infection, bacterial meningitis, and ototoxic medications such as those used to treat tuberculosis, pneumonia, or malaria. <ref>{{Cite journal|last=Paradowska|first=Edyta|last2=Jabłońska|first2=Agnieszka|last3=Studzińska|first3=Mirosława|last4=Kasztelewicz|first4=Beata|last5=Zawilińska|first5=Barbara|last6=Wiśniewska‐Ligier|first6=Małgorzata|last7=Dzierżanowska‐Fangrat|first7=Katarzyna|last8=Woźniakowska‐Gęsicka|first8=Teresa|last9=Kosz‐Vnenchak|first9=Magdalena|date=2014-08|title=Cytomegalovirus glycoprotein H genotype distribution and the relationship with hearing loss in children|url=https://onlinelibrary.wiley.com/doi/10.1002/jmv.23906|journal=Journal of Medical Virology|language=en|volume=86|issue=8|pages=1421–1427|doi=10.1002/jmv.23906|issn=0146-6615}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> The prevalence of disabling hearing loss in Sub-Saharan Africa is estimated to be 4.55% for all ages. The World Health Organization (WHO) classifies hearing loss as Mild (26-40 dBHL), Moderate (41-60 dBHL), Severe (61-80dBHL) and Profound (≥81dBHL). Hearing loss is a hidden disability as it is invisible and therefore often goes undetected and untreated. The prevalence of permanent hearing loss in developing nations is between 3-6/1000 in newborn babies while in developed countries the range is between 1-3/1000. <ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> <ref>{{Cite journal|last=Simões|first=Eric A. F.|last2=Kiio|first2=Francis|last3=Carosone-Link|first3=Phyllis J.|last4=Ndegwa|first4=Serah N.|last5=Ayugi|first5=John|last6=Macharia|first6=Isaac M.|date=2016-12|title=Otitis Media and Its Sequelae in Kenyan Schoolchildren|url=https://academic.oup.com/jpids/article-lookup/doi/10.1093/jpids/piv038|journal=Journal of the Pediatric Infectious Diseases Society|language=en|volume=5|issue=4|pages=375–384|doi=10.1093/jpids/piv038|issn=2048-7193|pmc=5181359|pmid=26407271}}</ref> {{HTitle|Information About Audiology}} There is limited information available specifically about the practice of audiology in Kenya. A 2018 study conducted in Malindi, Kenya, demonstrated the feasibility of using non-otolaryngology-based hearing screening programs. Community health workers and nursing staff successfully performed audiograms and otoscopic endoscopy using mobile technology, indicating potential for expanding audiology services through community-based initiatives.<ref>{{Cite journal|last=Jayawardena|first=Asitha D. L.|last2=Kahue|first2=Charissa N.|last3=Cummins|first3=Samantha M.|last4=Netterville|first4=James L.|date=2018-01|title=Expanding the Capacity of Otolaryngologists in Kenya through Mobile Technology|url=https://aao-hnsfjournals.onlinelibrary.wiley.com/doi/10.1177/2473974X18766824|journal=OTO Open|language=en|volume=2|issue=1|doi=10.1177/2473974X18766824|issn=2473-974X}}</ref> In 2024 a joint Nairobi-and Seattle-based group examined and published strategies for developing national EHC programs in Kenya. Recommendations include leveraging existing health policies, strengthening referral pathways, training healthcare workers, and engaging local partners for phased scale-up. These strategies aim to improve access to screening, diagnostics, management, and developmental support for hearing loss.<ref>{{Cite journal|last=Ndegwa|first=Serah|last2=Pavlik|first2=Michelle|last3=Gallagher|first3=Emily R.|last4=King’e|first4=Maureen|last5=Bocha|first5=Manaseh|last6=Mokoh|first6=Lilian Wairimu|last7=Macharia|first7=Isaac|last8=Stringer|first8=Paige|last9=Njuguna|first9=Irene|date=2024-02-05|title=Hearing Loss Detection and Early Intervention Strategies in Kenya|url=https://annalsofglobalhealth.org/articles/10.5334/aogh.4336/|journal=Annals of Global Health|language=en|volume=90|issue=1|doi=10.5334/aogh.4336|issn=2214-9996|pmc=10854416|pmid=38344005}}</ref> {{HTitle|Scope of Practice and Licensing}} As of 2025, there is one training program for clinical audiologists and speech-language pathologists through the University of Nairobi. The degree is a Bachelor is Science in Audiology, Speech, and Language Pathology (B.ASLP). The program is 4 years and the final year includes a clinical internship. <ref>https://surgery.uonbi.ac.ke/programs/bachelor-science-audiology-speech-and-language-pathology-baslp</ref> {{HTitle|Professional and Regulatory Bodies}} *The Kenya Ear Nose and Throat Society (KENTS) founded in 1988, represents otorhinolaryngologists, ENT Clinical Officers, Audiologists, and ENT nursing officers. It provides continuing medical education and professional development for its members. *The Association of Speech and Language Therapists Kenya works with audiologists on hearing care initiatives and is part of the national Ear and Hearing Care strategic planning team. * The Kenya Medical Practitioners and Dentists Board (KMPDB) is the statutory body that regulates audiology centres and hearing aid practices in Kenya. It board regulates the training and practice of medicine, dentistry, and community oral health, and audiology. Audiology centers must be registered with them to operate legally. {{HTitle|Ongoing audiology research}} Research in Kenya is focusing on addressing hearing care through early detection, community involvement, policy development, and innovative technologies and treatment approaches. Continued efforts are needed to overcome existing barriers and ensure comprehensive, accessible hearing care for all. {{HTitle|Challenges, Opportunities and Notes}} Challenges such as lack of equipment, training, and follow-up systems are significant barriers to effective hearing care. The majority of audiology services in Kenya are centralized in the capital, Nairobi. Families living outside of the capital are generally underserved. Additionally, there is limited access to services due to a small number of providers. These barriers lead to low awareness, delayed diagnosis and gaps in hearing care in Kenya. <ref> https://childrenwithhearingloss.org/kenya-program/</ref> The improvement of data management and follow-up systems, and leveraging local partnerships for a phased scale-up are opportunities that could significantly improve the provision of hearing services. {{HTitle|Audiology Charities}} * Kenya Society for Deaf Children (KSDC) * Deaf Empowerment Society of Kenya (DESK) * Kentalis International Foundation] * Deaf Child Worldwide] (international organization working in Kenya) {{HTitle|References}} {{reflist}} === External links=== '' Professional associations'' * Kenya Ear Nose and Throat Society ([https://kenyaentsociety.or.ke/ KENTS]) * Association of Speech and Language Therapists Kenya ([https://asltk.blogspot.com/ ASLTK]) * Kenya Medical Practitioners and Dentists Board ([https://kmpdc.go.ke/ KMPDB]) '' Charities'' * Kenya Society for Deaf Children ([https://deafchildrensociety-kenya.org/ KSDC]) * Deaf Empowerment Society of Kenya ([https://deskkenya.org/ DESK][https://www.kentalis.com/]) * [https://www.kentalis.com/ Kentalis International Foundation] * Deaf Child Worldwide (international organization working in Kenya, [https://www.ndcs.org.uk/deaf-child-worldwide/ NDCS]) {{:Global Audiology/Authors-1|Gabriella Green|https://www.linkedin.com/in/gabriellagreen206/}} [[Category:Audiology]] [[Category:Kenya]] i3ikjzouykes683tmjpri3km2vcz0yv 0 329686 2834950 2834181 2026-09-29T01:58:54Z Jtneill 10242 Move topic to 2026 2834950 wikitext text/x-wiki {{In progress}} ==Motivation== # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|User Name}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Motivating virtual teams/]] - How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does it affect behaviour? {{ME-By|User Name}} # [[/Veteran identity reconstruction and motivation/]] - How does identity reconstruction influence motivation during military-to-civilian transition? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How does unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}} ==Emotion== # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion regulation ability and strategy/]] - How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}} # [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}} # [[/Identity exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}} # [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Perfectionism and athlete mental health/]] - How does perfectionism affect athlete mental health? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|User Name}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}} ==Motivation and emotion== # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}} # [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}} [[Category:Motivation and emotion/Book/2027]] 5dcup3jz13vamq909d3fb0mbug9wo6o 0 330073 2835017 2833701 2026-09-29T09:55:37Z KB3250298 3105557 /* What exercise is best to help regulate emotions? */ 2835017 wikitext text/x-wiki {{title|Emotion regulation through exercise:<br>How do people use exercise to regulate their emotional states?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}}'''Case study''' [[File:Laptop and girl biting pencil-pixabay.jpg|thumb|200px|'''Figure 1'''. Sarah feeling very unregulated with too much going on in her life.]] Sarah stayed up late last night studying as she has three assignments due in the same week. When her alarm goes off at 6am for work she feels exhausted so crawls to the shower to wake herself up. Sarahs house mate bangs on the door yelling to hurry up as they need to use the bathroom. Sarah is already sick of this day but gets dressed and rushes off to her part-time work at a before school care center. Her boss grumbles at her for being 10 minutes late and the children are complaining and fighting about the breakfast they served today. Sarah pushes though her shift, and when it's over, she gets back to her car and sits in it for a moment, She feels tired, grumpy and so frustrated with so many people right now she feels like she might explode. Sarah knows she should go for a run because she hasn't exercised all week, but she is just too tired. Maybe she will get Macca's on the way home and go to her couch to scroll on her phone while she is peacefully at home by herself. What should Sarah do to help regulate her rising emotions: Go for a run or home to scroll? {{RoundBoxBottom}} [[Emotion|Emotions]] are a part of what makes us human and affect us all differently (Izard, 2013). Emotions effect the entirety of a person including their body, perception, cognition, their actions, personality and even their relationships (Izard, 2013). Emotions alert us to sensory information that needs our attention to keep us safe and healthy mentally and physically and steers our actions in these times. Although emotions are very helpful and necessary, they can also have negative impact when they become maladaptive or last too long. This is when we need to regulate our emotions. Emotion regulation is a helpful and important process for humans to function effectively day to day (''Emotion regulation: Conceptual foundations'', 2007). [[File:Plutchik-wheel.svg|thumb|'''Figure 2'''. Plutchik's wheel of emotions {{ic|Explain relevance to the topic}}]] Emotions can be negative or positive to a person and can be triggered by a range of situations in our lives, some we can't control, but some we can. So what we do to keep our emotions from becoming too much? {{RoundBoxTop|theme=3}}'''Focus questions''' * What is emotion regulation? * How can exercise help regulate emotions? * What exercise is best to help regulate emotions? {{RoundBoxBottom}} == What is emotion regulation? == [https://www.paulekman.com/universal-emotions/ Universal emotions] were identified by Paul Ekman, that we ''all'' feel, despite we speak different languages, live in different place regional, have different cultural, and ethnic differences (Ekman et al., 1999). He included happiness/enjoyment, anger, fear, disgust, sadness, surprise and then later added contempt. Plutchik went on to create a wheel of emotions to help visually explain these different emotions and show further breakdown ("The emotions," 1991). Emotions can also become dysregulated. [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional dysregulation] is when your emotions feel too big and too strong and a person loses the ability to control their behaviour. Emotional dysregulation is the inability to be aware of, accept, regulate, and modify emotional reactions and subsequent behaviours (Gratz & Roemer, 2004). It can look and feel like irritability, anger, mood swings, outbursts, feeling numb or detached, shutting down. Or in more extreme cases aggressive behaviors, stained relationships and connections and verbals shouting and crying<ref>{{Cite web|url=https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation|title=Cleveland Clinic: Emotional-dysregulation}}</ref>. The goal of emotion regulation is often explained as down-regulating (reducing) the sensation and/or duration of negative emotions such as feeling sad, angry or anxious (Gross et al., 2006). People can also try to up-regulate and increase the intensity or duration of positive emotions (Quoidbach et al., 2010). Emotion regulation is a very important skill of which children typically struggle with and some this can follow into adolescents and adulthood for some. == How can exercise help to regulate emotions? == Exercise is an essential part of humans everyday lives. It's recommended by [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians The Department of Health, Disability and Aging to move most days].  {{RoundBoxTop|theme=2}}'''Department of Health, Disability and Aging recommended physical activity for adults''' -Moderate to vigorous physical activities for at least 30 minutes on most days. -Muscle-strengthening activities at least twice per week. -Functional activities targeting mobility, balance, and coordination on 3 or more days per week. -Several hours of light-intensity physical activity daily. {{RoundBoxBottom}} === Physiological effects === Exercise positively attributes to well emotional regulation by changing brain chemistry, strengthening brain networks, and reducing stress hormones. Exercise has been proven to reduce cortisol levels and increase dopamine and other endorphins which are hormones responsible for the feelings of calm and comfort. Simply said, exercise puts the brain in a state where it is better able to regulate emotions! (reference) * Less cortisol and more dopamine mean emotional regulation is much easier. (reference) * '''Improve sleep:''' Exercise promotes better sleep as physical exertion regulates stress, lowers stress hormones and reduces muscle tension. === Psychological effects === Psychological benefits of exercise have continued to be support in many ways such as: * '''Mindfulness:''' Exercise can give people oppotunities to be present in their movements and have a healthy break from negative of challenging mental strains. * '''Connection:''' Exercise can be a fantastic way to connect with others and creat social support networks. Walking with friends or family is a perfect way to end a tough day. * '''Building self-efficacy and self-belief:''' Setting and reaching fitness goals can build personal xonfidence, build resilitience and give you a sense of control over your daily structure. Building self-efficacy and self-esteem through exercise is an important and healthy way to cope with daily challenges. * '''Improved mental health:''' Exercise has been proven to improve mental health. == Psychological theories == Many psychological theories can be applied to exercise and emotional regulation however here we will cover the process model of emotion regulation, [[w:Self determination theory|self-determination theory]] (SDT), [[cognitive evaluation theory]] (CET), [[drive theory|drive theory/ Conservation of Resources (COR) Theory]] and xxxxxxx. === Process model of emotion regulation (Gross, 19XX) === Gross's (19XX) Process model of emotion regulation in one of the most influential frameworks for understanding how emotions arise and how we can regulate them. === Self-determination theory (Deci & Ryan, 1985) === The self-determination theory focuses on fulfilling three core psychological needs, autonomy, competency and relatedness (need reference??). The SDT can be applied to exercise thanks to the metadata review by Teixeira et al., 2012. # <u>Autonomy:</u> Studies have consistently supported SDT aspects showing a positive relationship between more autonomous types of motivation and exercise. Autonomous motivation in terms of exercise, means excercising because you ''want'' to, not because you have to. Autonmous motvation towards exercise is a key mindset for consistency to develop. Individuals need to feel in control of their own actions, choices and so should choose their own exercise that suits them personally. The individual should be in charge of implementing structures and routines that work for them. # <u>Competence:</u> Competence satisfaction predicts better exercise participation (Teixeira et al., 2012). People are more likely to exercise if they experience competence satisfaction and feel like they are good at what they are doing. Individuals need to feel competent, can master challenging situations and feel as though they are improving and feel able to reach their own goals. This can be applied to fitness goals. # <u>Relatedness:</u> Relatedness is the need to experience a sense of belonging, connection, and care for others. People want to feel accepted, included and a part of a group. Relatedness is the desire to feel connected and interpersonally involved in warm, caring, reciprocal relationships. Exercise is a fantastic way to not just get the physical benefit of moving your body, but also the social and connection of doing so, alongside others with a common health goal. Joining a run club, aqua aerobics or tai chi class can help make social connections whilst moving your body. ==== Cognitive evaluation theory (Deci & Ryan, 1985) ==== Cognitive evaluation theory is a sub-theory of self-determination theory that explains how external events influence intrinsic motivation by affecting individual <u>perceived</u> competence and autonomy. Edward L. Deci and Richard M. Ryan develop CET in 1985, with the focus on the interplay between intrinsic and extrinsic motivation. In short, what drives human motiovation. CET examines how external factors—such as rewards, feedback, deadlines, or social pressures—can either enhance or undermine a person’s natural drive to engage in an activity for its own sake. The theory emphasizes that intrinsic motivation is not fixed but highly sensitive to environmental cues. In an sports or exercise context, individuals or coaches can use CET principles to encourage skill development and enjoyment rather than relying solely on external rewards such as first place trophies. Types of motivation from cognitive evaluation theory: * '''Intrinsic motivation:''' Doing an activity purely for its satisfaction, enjoyment, or interest predicts better exercise participation of long-term exercise adherence. A trend towards identified regulation (exercising because you see value in doing so) was found to predict participation more strongly than intrinsic motivation in the short term. * '''Extrinsic motivation:''' Doing an activity for external rewards, pressures, or avoidance of consequences, which can range from controlled (forced) to autonomous (internalised values). This is a good way to start exercising however intrinsic motives are better for long-term exercise habits. Extrinsic motives include winning a medal, gaining respect and praise or losing weight. * '''Amotivation:''' Amotivation is complete lack of intent or drive to take action. Three aspects attribute to feeling amotivative: Low competence, low autonomy, and low relatedness. The amotivated state is a predictor of poor performance and poor functioning. === Drive theory (Hull, 1943) === The drive theory proposes that humans are motivated to reduces pressures and stress and discomfort. Hull (1943) suggests that as a individuals arousal level increase, performance will also increase. Alternatively, if an individual's arousal levels decrease, then the performance would also decrease. Although this is not a primary motivational theory in the modern time, it still gives a good overview on what modern individuals often do not choose to exercise in time of emotional dysregulation. People are unconsciously looking to preserve energy and reduce stresses in the short term. What makes drive theory a learning theory, not just a motivation theory, is the feedback loop. When a particular behavior successfully reduces a drive, the connection between the situation and that behavior gets stronger. Hull called this “habit strength.” This is another point that is related for modern day people, we need to consciously turn to strengthen healthy habits and weaken maladaptive automatic responses.{{RoundBoxTop|theme=3}}'''So how can the self-determination theory be applied to Sarah?''' Autonomy: Sarah hates running, but she loves dancing. Zumba classes at 7:30pm twice a week might suit Sarah better after she has a quick tea and a rest straight after work. Competency: Sarah hasn't even done dance classes before, but she does feel confident in showing off her moves. Zumba classes could motivate her to move her body while building more range in her dance move repertoire. Relatedness: Sarah is a very social person and some of her friends might even like to join the Zumba classes too. {{RoundBoxBottom}} # == What exercise is best to help regulate emotions? == Australian adults should be active daily, undertaking a variety of physical activities at different intensities, which can include<ref>{{Cite web|url=https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians/recommendations-for-adults-18-to-64-years?language=en|title=Physical Activity Recommendations for Australian adults|date=|last=Australian Government}}</ref>: * moderate- to vigorous-intensity physical activities for 30 minutes or more on most days * muscle-strengthening activities on 2 or more days per week * functional activities targeting mobility, balance, and coordination on 3 or more days per week * several hours of light-intensity physical activity daily. Any movement is beneficial to individuals but to regulate your emotions the most ideal exercise looks like this: Anaerobic exercise (also known as resistance training) has benefits on emotion regulation such as....... Aerobic exercise<ref>{{Cite web|url=https://my.clevelandclinic.org/health/articles/7050-aerobic-exercise|title=Aerobic Exercise|last=Cleveland Clinic}}</ref> is when you continuously use oxygen as your fuel. Aerobic exercise promotes emotion regulation (Wang et al., 2024). And what is more important to note is that there are <u>immediate</u> results on mood. * {| class="wikitable" |+'''Table 1:''' Optimal exercise for each age category, with supporting studies. !Age !Exercise !Intensity/ Frequency !Study |- |Children |A variety of exercise such as running, soccer, dancing, tips, games and anything else that sustains interest and fun of the child. |60 minutes daily minimum Moderate–vigorous |Chaput et al., 2020 |- |Adolesence |A range of aerobic and strength exercise which can include sports, gym, HIT sessions, weights or running. |60 minutes daily Moderate–vigorous |Salvan et al., 2021 Parrish et al., 2020 |- |Adults |Walking, riding, swimming, running, zumba or sports. Begin with short, low-intensity walking and gradually increase duration and intensity |30 minutes or more, most days Moderate–vigorous strength twice a week |Ligeza et al., 2023, Mizzi et al., 2022 |- |Older Adults |Walking, exercise bike or water aerobics |Set depending on exercise ability and history |Erickson et al., 2011 |} ;Test yourself<quiz display="simple"> {High intensity running is best for older adults to help manager their emotions: |type="()"} + True - False {Children should take part in more frequent exercise than adults and it should be more varied: |type="()"} - True + False </quiz> ** ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== In conclusion, any physical activity is beneficial in times of emotional dysregulation. Using exercise as a preventative mechanism is also important. '''Useful takeaways:''' * One session of aerobic exercise can significantly boost positive mood and alleviate stress responses. It is recommended for use as an immediate mood regulation strategy. * Any movement can be used such as walking, dancing or sport. *Recommended length: 150 to 330 words {{tip|Key messages * Start by moving your body for 30 minutes in any way you enjoy. Positive effects on mood starts immediately. What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Erickson, K. I., Voss, M. W., Prakash, R. S., Basak, C., Szabo, A., Chaddock, L., Kim, J. S., Heo, S., Alves, H., White, S. M., Wojcicki, T. R., Mailey, E., Vieira, V. J., Martin, S. A., Pence, B. D., Woods, J. A., McAuley, E., & Kramer, A. F. (2011). Exercise training increases size of hippocampus and improves memory. Proceedings of the National Academy of Sciences, 108(7), 3017-3022. https://doi.org/10.1073/pnas.1015950108 Gratz, K. L., & Roemer, L. (2004). Multidimensional Assessment of Emotion Regulation and Dysregulation: Development, Factor Structure, and Initial Validation of the Difficulties in Emotion Regulation Scale. Journal of Psychopathology and Behavioral Assessment, 26(1), 41-54. https://doi.org/10.1023/B:JOBA.0000007455.08539.94 Izard, C. E. (2013). Human emotions. Springer Science & Business Media. Ligeza, T. S., Maciejczyk, M., Wyczesany, M., & Junghofer, M. (2023). The effects of a single aerobic exercise session on mood and neural emotional reactivity in depressed and healthy young adults: A late positive potential study. Psychophysiology, 60(1), e14137. https://doi.org/https://doi.org/10.1111/psyp.14137 Salvan, P., Wassenaar, T., Wheatley, C., Beale, N., Cottaar, M., Papp, D., Bastiani, M., Fitzgibbon, S., Duff, E., Andersson, J., Winkler, A. M., Douaud, G., Nichols, T. E., Smith, S., Dawes, H., & Johansen-Berg, H. (2021). Multimodal Imaging Brain Markers in Early Adolescence Are Linked with a Physically Active Lifestyle. The Journal of Neuroscience, 41(5), 1092. https://doi.org/10.1523/JNEUROSCI.1260-20.2020 Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., & Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: a systematic review. International journal of behavioral nutrition and physical activity, 9(1), 78. Wang, X., Liu, T., Jin, X., & Zhou, C. (2024). Aerobic exercise promotes emotion regulation: a narrative review. Experimental Brain Research, 242(4), 783-796. https://doi.org/https://doi.org/10.1007/s00221-024-06791-1 }} ==External links== External relevant resources: * [https://www.paulekman.com/universal-emotions/ Universal Emotions | What are Emotions? | Paul Ekman Group] * [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional Dysregulation: What It Is, Causes & Treatment] * [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians 24-hour movement guidelines for all Australians | Australian Government Department of Health, Disability and Ageing] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] ==See also== * [[Motivation and emotion/Book/2025/Emotion regulation through exercise|Emotion regulation through exercise]] (Book chapter, 2025) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional self-regulation]] [[Category:Motivation and emotion/Book/Exercise]] <noinclude> <hr> __TOC__ [[Category:Motivation and emotion/Book]] gf68iblm2f9y9h0tkvy5dp1iuq34q8n 2835023 2835017 2026-09-29T10:37:56Z KB3250298 3105557 /* Physiological effects */ 2835023 wikitext text/x-wiki {{title|Emotion regulation through exercise:<br>How do people use exercise to regulate their emotional states?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}}'''Case study''' [[File:Laptop and girl biting pencil-pixabay.jpg|thumb|200px|'''Figure 1'''. Sarah feeling very unregulated with too much going on in her life.]] Sarah stayed up late last night studying as she has three assignments due in the same week. When her alarm goes off at 6am for work she feels exhausted so crawls to the shower to wake herself up. Sarahs house mate bangs on the door yelling to hurry up as they need to use the bathroom. Sarah is already sick of this day but gets dressed and rushes off to her part-time work at a before school care center. Her boss grumbles at her for being 10 minutes late and the children are complaining and fighting about the breakfast they served today. Sarah pushes though her shift, and when it's over, she gets back to her car and sits in it for a moment, She feels tired, grumpy and so frustrated with so many people right now she feels like she might explode. Sarah knows she should go for a run because she hasn't exercised all week, but she is just too tired. Maybe she will get Macca's on the way home and go to her couch to scroll on her phone while she is peacefully at home by herself. What should Sarah do to help regulate her rising emotions: Go for a run or home to scroll? {{RoundBoxBottom}} [[Emotion|Emotions]] are a part of what makes us human and affect us all differently (Izard, 2013). Emotions effect the entirety of a person including their body, perception, cognition, their actions, personality and even their relationships (Izard, 2013). Emotions alert us to sensory information that needs our attention to keep us safe and healthy mentally and physically and steers our actions in these times. Although emotions are very helpful and necessary, they can also have negative impact when they become maladaptive or last too long. This is when we need to regulate our emotions. Emotion regulation is a helpful and important process for humans to function effectively day to day (''Emotion regulation: Conceptual foundations'', 2007). [[File:Plutchik-wheel.svg|thumb|'''Figure 2'''. Plutchik's wheel of emotions {{ic|Explain relevance to the topic}}]] Emotions can be negative or positive to a person and can be triggered by a range of situations in our lives, some we can't control, but some we can. So what we do to keep our emotions from becoming too much? {{RoundBoxTop|theme=3}}'''Focus questions''' * What is emotion regulation? * How can exercise help regulate emotions? * What exercise is best to help regulate emotions? {{RoundBoxBottom}} == What is emotion regulation? == [https://www.paulekman.com/universal-emotions/ Universal emotions] were identified by Paul Ekman, that we ''all'' feel, despite we speak different languages, live in different place regional, have different cultural, and ethnic differences (Ekman et al., 1999). He included happiness/enjoyment, anger, fear, disgust, sadness, surprise and then later added contempt. Plutchik went on to create a wheel of emotions to help visually explain these different emotions and show further breakdown ("The emotions," 1991). Emotions can also become dysregulated. [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional dysregulation] is when your emotions feel too big and too strong and a person loses the ability to control their behaviour. Emotional dysregulation is the inability to be aware of, accept, regulate, and modify emotional reactions and subsequent behaviours (Gratz & Roemer, 2004). It can look and feel like irritability, anger, mood swings, outbursts, feeling numb or detached, shutting down. Or in more extreme cases aggressive behaviors, stained relationships and connections and verbals shouting and crying<ref>{{Cite web|url=https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation|title=Cleveland Clinic: Emotional-dysregulation}}</ref>. The goal of emotion regulation is often explained as down-regulating (reducing) the sensation and/or duration of negative emotions such as feeling sad, angry or anxious (Gross et al., 2006). People can also try to up-regulate and increase the intensity or duration of positive emotions (Quoidbach et al., 2010). Emotion regulation is a very important skill of which children typically struggle with and some this can follow into adolescents and adulthood for some. == How can exercise help to regulate emotions? == Exercise is an essential part of humans everyday lives. It's recommended by [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians The Department of Health, Disability and Aging to move most days].  {{RoundBoxTop|theme=2}}'''Department of Health, Disability and Aging recommended physical activity for adults''' -Moderate to vigorous physical activities for at least 30 minutes on most days. -Muscle-strengthening activities at least twice per week. -Functional activities targeting mobility, balance, and coordination on 3 or more days per week. -Several hours of light-intensity physical activity daily. {{RoundBoxBottom}} === Physiological effects === Exercise positively attributes to well emotional regulation by changing brain chemistry, strengthening brain networks, and reducing stress hormones. Exercise has been proven to reduce cortisol levels and increase dopamine and other endorphins which are hormones responsible for the feelings of calm and comfort. Simply said, exercise puts the brain in a state where it is better able to regulate emotions! (reference) * '''Reduces cortisol and increases dopamine''': Meaning emotional regulation is much easier. (reference) * '''Improve sleep:''' Exercise promotes better sleep as physical exertion regulates stress, lowers stress hormones and reduces muscle tension. === Psychological effects === Psychological benefits of exercise have continued to be support in many ways such as: * '''Mindfulness:''' Exercise can give people oppotunities to be present in their movements and have a healthy break from negative of challenging mental strains. * '''Connection:''' Exercise can be a fantastic way to connect with others and creat social support networks. Walking with friends or family is a perfect way to end a tough day. * '''Building self-efficacy and self-belief:''' Setting and reaching fitness goals can build personal xonfidence, build resilitience and give you a sense of control over your daily structure. Building self-efficacy and self-esteem through exercise is an important and healthy way to cope with daily challenges. * '''Improved mental health:''' Exercise has been proven to improve mental health. == Psychological theories == Many psychological theories can be applied to exercise and emotional regulation however here we will cover the process model of emotion regulation, [[w:Self determination theory|self-determination theory]] (SDT), [[cognitive evaluation theory]] (CET), [[drive theory|drive theory/ Conservation of Resources (COR) Theory]] and xxxxxxx. === Process model of emotion regulation (Gross, 19XX) === Gross's (19XX) Process model of emotion regulation in one of the most influential frameworks for understanding how emotions arise and how we can regulate them. === Self-determination theory (Deci & Ryan, 1985) === The self-determination theory focuses on fulfilling three core psychological needs, autonomy, competency and relatedness (need reference??). The SDT can be applied to exercise thanks to the metadata review by Teixeira et al., 2012. # <u>Autonomy:</u> Studies have consistently supported SDT aspects showing a positive relationship between more autonomous types of motivation and exercise. Autonomous motivation in terms of exercise, means excercising because you ''want'' to, not because you have to. Autonmous motvation towards exercise is a key mindset for consistency to develop. Individuals need to feel in control of their own actions, choices and so should choose their own exercise that suits them personally. The individual should be in charge of implementing structures and routines that work for them. # <u>Competence:</u> Competence satisfaction predicts better exercise participation (Teixeira et al., 2012). People are more likely to exercise if they experience competence satisfaction and feel like they are good at what they are doing. Individuals need to feel competent, can master challenging situations and feel as though they are improving and feel able to reach their own goals. This can be applied to fitness goals. # <u>Relatedness:</u> Relatedness is the need to experience a sense of belonging, connection, and care for others. People want to feel accepted, included and a part of a group. Relatedness is the desire to feel connected and interpersonally involved in warm, caring, reciprocal relationships. Exercise is a fantastic way to not just get the physical benefit of moving your body, but also the social and connection of doing so, alongside others with a common health goal. Joining a run club, aqua aerobics or tai chi class can help make social connections whilst moving your body. ==== Cognitive evaluation theory (Deci & Ryan, 1985) ==== Cognitive evaluation theory is a sub-theory of self-determination theory that explains how external events influence intrinsic motivation by affecting individual <u>perceived</u> competence and autonomy. Edward L. Deci and Richard M. Ryan develop CET in 1985, with the focus on the interplay between intrinsic and extrinsic motivation. In short, what drives human motiovation. CET examines how external factors—such as rewards, feedback, deadlines, or social pressures—can either enhance or undermine a person’s natural drive to engage in an activity for its own sake. The theory emphasizes that intrinsic motivation is not fixed but highly sensitive to environmental cues. In an sports or exercise context, individuals or coaches can use CET principles to encourage skill development and enjoyment rather than relying solely on external rewards such as first place trophies. Types of motivation from cognitive evaluation theory: * '''Intrinsic motivation:''' Doing an activity purely for its satisfaction, enjoyment, or interest predicts better exercise participation of long-term exercise adherence. A trend towards identified regulation (exercising because you see value in doing so) was found to predict participation more strongly than intrinsic motivation in the short term. * '''Extrinsic motivation:''' Doing an activity for external rewards, pressures, or avoidance of consequences, which can range from controlled (forced) to autonomous (internalised values). This is a good way to start exercising however intrinsic motives are better for long-term exercise habits. Extrinsic motives include winning a medal, gaining respect and praise or losing weight. * '''Amotivation:''' Amotivation is complete lack of intent or drive to take action. Three aspects attribute to feeling amotivative: Low competence, low autonomy, and low relatedness. The amotivated state is a predictor of poor performance and poor functioning. === Drive theory (Hull, 1943) === The drive theory proposes that humans are motivated to reduces pressures and stress and discomfort. Hull (1943) suggests that as a individuals arousal level increase, performance will also increase. Alternatively, if an individual's arousal levels decrease, then the performance would also decrease. Although this is not a primary motivational theory in the modern time, it still gives a good overview on what modern individuals often do not choose to exercise in time of emotional dysregulation. People are unconsciously looking to preserve energy and reduce stresses in the short term. What makes drive theory a learning theory, not just a motivation theory, is the feedback loop. When a particular behavior successfully reduces a drive, the connection between the situation and that behavior gets stronger. Hull called this “habit strength.” This is another point that is related for modern day people, we need to consciously turn to strengthen healthy habits and weaken maladaptive automatic responses.{{RoundBoxTop|theme=3}}'''So how can the self-determination theory be applied to Sarah?''' Autonomy: Sarah hates running, but she loves dancing. Zumba classes at 7:30pm twice a week might suit Sarah better after she has a quick tea and a rest straight after work. Competency: Sarah hasn't even done dance classes before, but she does feel confident in showing off her moves. Zumba classes could motivate her to move her body while building more range in her dance move repertoire. Relatedness: Sarah is a very social person and some of her friends might even like to join the Zumba classes too. {{RoundBoxBottom}} # == What exercise is best to help regulate emotions? == Australian adults should be active daily, undertaking a variety of physical activities at different intensities, which can include<ref>{{Cite web|url=https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians/recommendations-for-adults-18-to-64-years?language=en|title=Physical Activity Recommendations for Australian adults|date=|last=Australian Government}}</ref>: * moderate- to vigorous-intensity physical activities for 30 minutes or more on most days * muscle-strengthening activities on 2 or more days per week * functional activities targeting mobility, balance, and coordination on 3 or more days per week * several hours of light-intensity physical activity daily. Any movement is beneficial to individuals but to regulate your emotions the most ideal exercise looks like this: Anaerobic exercise (also known as resistance training) has benefits on emotion regulation such as....... Aerobic exercise<ref>{{Cite web|url=https://my.clevelandclinic.org/health/articles/7050-aerobic-exercise|title=Aerobic Exercise|last=Cleveland Clinic}}</ref> is when you continuously use oxygen as your fuel. Aerobic exercise promotes emotion regulation (Wang et al., 2024). And what is more important to note is that there are <u>immediate</u> results on mood. * {| class="wikitable" |+'''Table 1:''' Optimal exercise for each age category, with supporting studies. !Age !Exercise !Intensity/ Frequency !Study |- |Children |A variety of exercise such as running, soccer, dancing, tips, games and anything else that sustains interest and fun of the child. |60 minutes daily minimum Moderate–vigorous |Chaput et al., 2020 |- |Adolesence |A range of aerobic and strength exercise which can include sports, gym, HIT sessions, weights or running. |60 minutes daily Moderate–vigorous |Salvan et al., 2021 Parrish et al., 2020 |- |Adults |Walking, riding, swimming, running, zumba or sports. Begin with short, low-intensity walking and gradually increase duration and intensity |30 minutes or more, most days Moderate–vigorous strength twice a week |Ligeza et al., 2023, Mizzi et al., 2022 |- |Older Adults |Walking, exercise bike or water aerobics |Set depending on exercise ability and history |Erickson et al., 2011 |} ;Test yourself<quiz display="simple"> {High intensity running is best for older adults to help manager their emotions: |type="()"} + True - False {Children should take part in more frequent exercise than adults and it should be more varied: |type="()"} - True + False </quiz> ** ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== In conclusion, any physical activity is beneficial in times of emotional dysregulation. Using exercise as a preventative mechanism is also important. '''Useful takeaways:''' * One session of aerobic exercise can significantly boost positive mood and alleviate stress responses. It is recommended for use as an immediate mood regulation strategy. * Any movement can be used such as walking, dancing or sport. *Recommended length: 150 to 330 words {{tip|Key messages * Start by moving your body for 30 minutes in any way you enjoy. Positive effects on mood starts immediately. What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Erickson, K. I., Voss, M. W., Prakash, R. S., Basak, C., Szabo, A., Chaddock, L., Kim, J. S., Heo, S., Alves, H., White, S. M., Wojcicki, T. R., Mailey, E., Vieira, V. J., Martin, S. A., Pence, B. D., Woods, J. A., McAuley, E., & Kramer, A. F. (2011). Exercise training increases size of hippocampus and improves memory. Proceedings of the National Academy of Sciences, 108(7), 3017-3022. https://doi.org/10.1073/pnas.1015950108 Gratz, K. L., & Roemer, L. (2004). Multidimensional Assessment of Emotion Regulation and Dysregulation: Development, Factor Structure, and Initial Validation of the Difficulties in Emotion Regulation Scale. Journal of Psychopathology and Behavioral Assessment, 26(1), 41-54. https://doi.org/10.1023/B:JOBA.0000007455.08539.94 Izard, C. E. (2013). Human emotions. Springer Science & Business Media. Ligeza, T. S., Maciejczyk, M., Wyczesany, M., & Junghofer, M. (2023). The effects of a single aerobic exercise session on mood and neural emotional reactivity in depressed and healthy young adults: A late positive potential study. Psychophysiology, 60(1), e14137. https://doi.org/https://doi.org/10.1111/psyp.14137 Salvan, P., Wassenaar, T., Wheatley, C., Beale, N., Cottaar, M., Papp, D., Bastiani, M., Fitzgibbon, S., Duff, E., Andersson, J., Winkler, A. M., Douaud, G., Nichols, T. E., Smith, S., Dawes, H., & Johansen-Berg, H. (2021). Multimodal Imaging Brain Markers in Early Adolescence Are Linked with a Physically Active Lifestyle. The Journal of Neuroscience, 41(5), 1092. https://doi.org/10.1523/JNEUROSCI.1260-20.2020 Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., & Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: a systematic review. International journal of behavioral nutrition and physical activity, 9(1), 78. Wang, X., Liu, T., Jin, X., & Zhou, C. (2024). Aerobic exercise promotes emotion regulation: a narrative review. Experimental Brain Research, 242(4), 783-796. https://doi.org/https://doi.org/10.1007/s00221-024-06791-1 }} ==External links== External relevant resources: * [https://www.paulekman.com/universal-emotions/ Universal Emotions | What are Emotions? | Paul Ekman Group] * [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional Dysregulation: What It Is, Causes & Treatment] * [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians 24-hour movement guidelines for all Australians | Australian Government Department of Health, Disability and Ageing] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] ==See also== * [[Motivation and emotion/Book/2025/Emotion regulation through exercise|Emotion regulation through exercise]] (Book chapter, 2025) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional self-regulation]] [[Category:Motivation and emotion/Book/Exercise]] <noinclude> <hr> __TOC__ [[Category:Motivation and emotion/Book]] idxhwy0ekxgdgxtczkera4pxttde9hf 2835026 2835023 2026-09-29T10:41:10Z KB3250298 3105557 /* Conclusion */ 2835026 wikitext text/x-wiki {{title|Emotion regulation through exercise:<br>How do people use exercise to regulate their emotional states?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}}'''Case study''' [[File:Laptop and girl biting pencil-pixabay.jpg|thumb|200px|'''Figure 1'''. Sarah feeling very unregulated with too much going on in her life.]] Sarah stayed up late last night studying as she has three assignments due in the same week. When her alarm goes off at 6am for work she feels exhausted so crawls to the shower to wake herself up. Sarahs house mate bangs on the door yelling to hurry up as they need to use the bathroom. Sarah is already sick of this day but gets dressed and rushes off to her part-time work at a before school care center. Her boss grumbles at her for being 10 minutes late and the children are complaining and fighting about the breakfast they served today. Sarah pushes though her shift, and when it's over, she gets back to her car and sits in it for a moment, She feels tired, grumpy and so frustrated with so many people right now she feels like she might explode. Sarah knows she should go for a run because she hasn't exercised all week, but she is just too tired. Maybe she will get Macca's on the way home and go to her couch to scroll on her phone while she is peacefully at home by herself. What should Sarah do to help regulate her rising emotions: Go for a run or home to scroll? {{RoundBoxBottom}} [[Emotion|Emotions]] are a part of what makes us human and affect us all differently (Izard, 2013). Emotions effect the entirety of a person including their body, perception, cognition, their actions, personality and even their relationships (Izard, 2013). Emotions alert us to sensory information that needs our attention to keep us safe and healthy mentally and physically and steers our actions in these times. Although emotions are very helpful and necessary, they can also have negative impact when they become maladaptive or last too long. This is when we need to regulate our emotions. Emotion regulation is a helpful and important process for humans to function effectively day to day (''Emotion regulation: Conceptual foundations'', 2007). [[File:Plutchik-wheel.svg|thumb|'''Figure 2'''. Plutchik's wheel of emotions {{ic|Explain relevance to the topic}}]] Emotions can be negative or positive to a person and can be triggered by a range of situations in our lives, some we can't control, but some we can. So what we do to keep our emotions from becoming too much? {{RoundBoxTop|theme=3}}'''Focus questions''' * What is emotion regulation? * How can exercise help regulate emotions? * What exercise is best to help regulate emotions? {{RoundBoxBottom}} == What is emotion regulation? == [https://www.paulekman.com/universal-emotions/ Universal emotions] were identified by Paul Ekman, that we ''all'' feel, despite we speak different languages, live in different place regional, have different cultural, and ethnic differences (Ekman et al., 1999). He included happiness/enjoyment, anger, fear, disgust, sadness, surprise and then later added contempt. Plutchik went on to create a wheel of emotions to help visually explain these different emotions and show further breakdown ("The emotions," 1991). Emotions can also become dysregulated. [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional dysregulation] is when your emotions feel too big and too strong and a person loses the ability to control their behaviour. Emotional dysregulation is the inability to be aware of, accept, regulate, and modify emotional reactions and subsequent behaviours (Gratz & Roemer, 2004). It can look and feel like irritability, anger, mood swings, outbursts, feeling numb or detached, shutting down. Or in more extreme cases aggressive behaviors, stained relationships and connections and verbals shouting and crying<ref>{{Cite web|url=https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation|title=Cleveland Clinic: Emotional-dysregulation}}</ref>. The goal of emotion regulation is often explained as down-regulating (reducing) the sensation and/or duration of negative emotions such as feeling sad, angry or anxious (Gross et al., 2006). People can also try to up-regulate and increase the intensity or duration of positive emotions (Quoidbach et al., 2010). Emotion regulation is a very important skill of which children typically struggle with and some this can follow into adolescents and adulthood for some. == How can exercise help to regulate emotions? == Exercise is an essential part of humans everyday lives. It's recommended by [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians The Department of Health, Disability and Aging to move most days].  {{RoundBoxTop|theme=2}}'''Department of Health, Disability and Aging recommended physical activity for adults''' -Moderate to vigorous physical activities for at least 30 minutes on most days. -Muscle-strengthening activities at least twice per week. -Functional activities targeting mobility, balance, and coordination on 3 or more days per week. -Several hours of light-intensity physical activity daily. {{RoundBoxBottom}} === Physiological effects === Exercise positively attributes to well emotional regulation by changing brain chemistry, strengthening brain networks, and reducing stress hormones. Exercise has been proven to reduce cortisol levels and increase dopamine and other endorphins which are hormones responsible for the feelings of calm and comfort. Simply said, exercise puts the brain in a state where it is better able to regulate emotions! (reference) * '''Reduces cortisol and increases dopamine''': Meaning emotional regulation is much easier. (reference) * '''Improve sleep:''' Exercise promotes better sleep as physical exertion regulates stress, lowers stress hormones and reduces muscle tension. === Psychological effects === Psychological benefits of exercise have continued to be support in many ways such as: * '''Mindfulness:''' Exercise can give people oppotunities to be present in their movements and have a healthy break from negative of challenging mental strains. * '''Connection:''' Exercise can be a fantastic way to connect with others and creat social support networks. Walking with friends or family is a perfect way to end a tough day. * '''Building self-efficacy and self-belief:''' Setting and reaching fitness goals can build personal xonfidence, build resilitience and give you a sense of control over your daily structure. Building self-efficacy and self-esteem through exercise is an important and healthy way to cope with daily challenges. * '''Improved mental health:''' Exercise has been proven to improve mental health. == Psychological theories == Many psychological theories can be applied to exercise and emotional regulation however here we will cover the process model of emotion regulation, [[w:Self determination theory|self-determination theory]] (SDT), [[cognitive evaluation theory]] (CET), [[drive theory|drive theory/ Conservation of Resources (COR) Theory]] and xxxxxxx. === Process model of emotion regulation (Gross, 19XX) === Gross's (19XX) Process model of emotion regulation in one of the most influential frameworks for understanding how emotions arise and how we can regulate them. === Self-determination theory (Deci & Ryan, 1985) === The self-determination theory focuses on fulfilling three core psychological needs, autonomy, competency and relatedness (need reference??). The SDT can be applied to exercise thanks to the metadata review by Teixeira et al., 2012. # <u>Autonomy:</u> Studies have consistently supported SDT aspects showing a positive relationship between more autonomous types of motivation and exercise. Autonomous motivation in terms of exercise, means excercising because you ''want'' to, not because you have to. Autonmous motvation towards exercise is a key mindset for consistency to develop. Individuals need to feel in control of their own actions, choices and so should choose their own exercise that suits them personally. The individual should be in charge of implementing structures and routines that work for them. # <u>Competence:</u> Competence satisfaction predicts better exercise participation (Teixeira et al., 2012). People are more likely to exercise if they experience competence satisfaction and feel like they are good at what they are doing. Individuals need to feel competent, can master challenging situations and feel as though they are improving and feel able to reach their own goals. This can be applied to fitness goals. # <u>Relatedness:</u> Relatedness is the need to experience a sense of belonging, connection, and care for others. People want to feel accepted, included and a part of a group. Relatedness is the desire to feel connected and interpersonally involved in warm, caring, reciprocal relationships. Exercise is a fantastic way to not just get the physical benefit of moving your body, but also the social and connection of doing so, alongside others with a common health goal. Joining a run club, aqua aerobics or tai chi class can help make social connections whilst moving your body. ==== Cognitive evaluation theory (Deci & Ryan, 1985) ==== Cognitive evaluation theory is a sub-theory of self-determination theory that explains how external events influence intrinsic motivation by affecting individual <u>perceived</u> competence and autonomy. Edward L. Deci and Richard M. Ryan develop CET in 1985, with the focus on the interplay between intrinsic and extrinsic motivation. In short, what drives human motiovation. CET examines how external factors—such as rewards, feedback, deadlines, or social pressures—can either enhance or undermine a person’s natural drive to engage in an activity for its own sake. The theory emphasizes that intrinsic motivation is not fixed but highly sensitive to environmental cues. In an sports or exercise context, individuals or coaches can use CET principles to encourage skill development and enjoyment rather than relying solely on external rewards such as first place trophies. Types of motivation from cognitive evaluation theory: * '''Intrinsic motivation:''' Doing an activity purely for its satisfaction, enjoyment, or interest predicts better exercise participation of long-term exercise adherence. A trend towards identified regulation (exercising because you see value in doing so) was found to predict participation more strongly than intrinsic motivation in the short term. * '''Extrinsic motivation:''' Doing an activity for external rewards, pressures, or avoidance of consequences, which can range from controlled (forced) to autonomous (internalised values). This is a good way to start exercising however intrinsic motives are better for long-term exercise habits. Extrinsic motives include winning a medal, gaining respect and praise or losing weight. * '''Amotivation:''' Amotivation is complete lack of intent or drive to take action. Three aspects attribute to feeling amotivative: Low competence, low autonomy, and low relatedness. The amotivated state is a predictor of poor performance and poor functioning. === Drive theory (Hull, 1943) === The drive theory proposes that humans are motivated to reduces pressures and stress and discomfort. Hull (1943) suggests that as a individuals arousal level increase, performance will also increase. Alternatively, if an individual's arousal levels decrease, then the performance would also decrease. Although this is not a primary motivational theory in the modern time, it still gives a good overview on what modern individuals often do not choose to exercise in time of emotional dysregulation. People are unconsciously looking to preserve energy and reduce stresses in the short term. What makes drive theory a learning theory, not just a motivation theory, is the feedback loop. When a particular behavior successfully reduces a drive, the connection between the situation and that behavior gets stronger. Hull called this “habit strength.” This is another point that is related for modern day people, we need to consciously turn to strengthen healthy habits and weaken maladaptive automatic responses.{{RoundBoxTop|theme=3}}'''So how can the self-determination theory be applied to Sarah?''' Autonomy: Sarah hates running, but she loves dancing. Zumba classes at 7:30pm twice a week might suit Sarah better after she has a quick tea and a rest straight after work. Competency: Sarah hasn't even done dance classes before, but she does feel confident in showing off her moves. Zumba classes could motivate her to move her body while building more range in her dance move repertoire. Relatedness: Sarah is a very social person and some of her friends might even like to join the Zumba classes too. {{RoundBoxBottom}} # == What exercise is best to help regulate emotions? == Australian adults should be active daily, undertaking a variety of physical activities at different intensities, which can include<ref>{{Cite web|url=https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians/recommendations-for-adults-18-to-64-years?language=en|title=Physical Activity Recommendations for Australian adults|date=|last=Australian Government}}</ref>: * moderate- to vigorous-intensity physical activities for 30 minutes or more on most days * muscle-strengthening activities on 2 or more days per week * functional activities targeting mobility, balance, and coordination on 3 or more days per week * several hours of light-intensity physical activity daily. Any movement is beneficial to individuals but to regulate your emotions the most ideal exercise looks like this: Anaerobic exercise (also known as resistance training) has benefits on emotion regulation such as....... Aerobic exercise<ref>{{Cite web|url=https://my.clevelandclinic.org/health/articles/7050-aerobic-exercise|title=Aerobic Exercise|last=Cleveland Clinic}}</ref> is when you continuously use oxygen as your fuel. Aerobic exercise promotes emotion regulation (Wang et al., 2024). And what is more important to note is that there are <u>immediate</u> results on mood. * {| class="wikitable" |+'''Table 1:''' Optimal exercise for each age category, with supporting studies. !Age !Exercise !Intensity/ Frequency !Study |- |Children |A variety of exercise such as running, soccer, dancing, tips, games and anything else that sustains interest and fun of the child. |60 minutes daily minimum Moderate–vigorous |Chaput et al., 2020 |- |Adolesence |A range of aerobic and strength exercise which can include sports, gym, HIT sessions, weights or running. |60 minutes daily Moderate–vigorous |Salvan et al., 2021 Parrish et al., 2020 |- |Adults |Walking, riding, swimming, running, zumba or sports. Begin with short, low-intensity walking and gradually increase duration and intensity |30 minutes or more, most days Moderate–vigorous strength twice a week |Ligeza et al., 2023, Mizzi et al., 2022 |- |Older Adults |Walking, exercise bike or water aerobics |Set depending on exercise ability and history |Erickson et al., 2011 |} ;Test yourself<quiz display="simple"> {High intensity running is best for older adults to help manager their emotions: |type="()"} + True - False {Children should take part in more frequent exercise than adults and it should be more varied: |type="()"} - True + False </quiz> ** ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== In conclusion, physical activity has physical, and psychological benefits and is beneficial in times of emotional dysregulation. Using exercise as a preventative mechanism is also important. '''Useful takeaways:''' * One session of aerobic exercise can significantly boost positive mood and alleviate stress responses. It is recommended for use as an immediate mood regulation strategy. * Any movement can be used such as walking, dancing or sport. *Recommended length: 150 to 330 words {{tip|Key messages * Start by moving your body for 30 minutes in any way you enjoy. Positive effects on mood starts immediately. What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Erickson, K. I., Voss, M. W., Prakash, R. S., Basak, C., Szabo, A., Chaddock, L., Kim, J. S., Heo, S., Alves, H., White, S. M., Wojcicki, T. R., Mailey, E., Vieira, V. J., Martin, S. A., Pence, B. D., Woods, J. A., McAuley, E., & Kramer, A. F. (2011). Exercise training increases size of hippocampus and improves memory. Proceedings of the National Academy of Sciences, 108(7), 3017-3022. https://doi.org/10.1073/pnas.1015950108 Gratz, K. L., & Roemer, L. (2004). Multidimensional Assessment of Emotion Regulation and Dysregulation: Development, Factor Structure, and Initial Validation of the Difficulties in Emotion Regulation Scale. Journal of Psychopathology and Behavioral Assessment, 26(1), 41-54. https://doi.org/10.1023/B:JOBA.0000007455.08539.94 Izard, C. E. (2013). Human emotions. Springer Science & Business Media. Ligeza, T. S., Maciejczyk, M., Wyczesany, M., & Junghofer, M. (2023). The effects of a single aerobic exercise session on mood and neural emotional reactivity in depressed and healthy young adults: A late positive potential study. Psychophysiology, 60(1), e14137. https://doi.org/https://doi.org/10.1111/psyp.14137 Salvan, P., Wassenaar, T., Wheatley, C., Beale, N., Cottaar, M., Papp, D., Bastiani, M., Fitzgibbon, S., Duff, E., Andersson, J., Winkler, A. M., Douaud, G., Nichols, T. E., Smith, S., Dawes, H., & Johansen-Berg, H. (2021). Multimodal Imaging Brain Markers in Early Adolescence Are Linked with a Physically Active Lifestyle. The Journal of Neuroscience, 41(5), 1092. https://doi.org/10.1523/JNEUROSCI.1260-20.2020 Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., & Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: a systematic review. International journal of behavioral nutrition and physical activity, 9(1), 78. Wang, X., Liu, T., Jin, X., & Zhou, C. (2024). Aerobic exercise promotes emotion regulation: a narrative review. Experimental Brain Research, 242(4), 783-796. https://doi.org/https://doi.org/10.1007/s00221-024-06791-1 }} ==External links== External relevant resources: * [https://www.paulekman.com/universal-emotions/ Universal Emotions | What are Emotions? | Paul Ekman Group] * [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional Dysregulation: What It Is, Causes & Treatment] * [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians 24-hour movement guidelines for all Australians | Australian Government Department of Health, Disability and Ageing] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] ==See also== * [[Motivation and emotion/Book/2025/Emotion regulation through exercise|Emotion regulation through exercise]] (Book chapter, 2025) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional self-regulation]] [[Category:Motivation and emotion/Book/Exercise]] <noinclude> <hr> __TOC__ [[Category:Motivation and emotion/Book]] 87v858jqp5w3oz727izp6bkvbx055xs 2835027 2835026 2026-09-29T10:47:14Z KB3250298 3105557 /* Drive theory (Hull, 1943) */ 2835027 wikitext text/x-wiki {{title|Emotion regulation through exercise:<br>How do people use exercise to regulate their emotional states?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}}'''Case study''' [[File:Laptop and girl biting pencil-pixabay.jpg|thumb|200px|'''Figure 1'''. Sarah feeling very unregulated with too much going on in her life.]] Sarah stayed up late last night studying as she has three assignments due in the same week. When her alarm goes off at 6am for work she feels exhausted so crawls to the shower to wake herself up. Sarahs house mate bangs on the door yelling to hurry up as they need to use the bathroom. Sarah is already sick of this day but gets dressed and rushes off to her part-time work at a before school care center. Her boss grumbles at her for being 10 minutes late and the children are complaining and fighting about the breakfast they served today. Sarah pushes though her shift, and when it's over, she gets back to her car and sits in it for a moment, She feels tired, grumpy and so frustrated with so many people right now she feels like she might explode. Sarah knows she should go for a run because she hasn't exercised all week, but she is just too tired. Maybe she will get Macca's on the way home and go to her couch to scroll on her phone while she is peacefully at home by herself. What should Sarah do to help regulate her rising emotions: Go for a run or home to scroll? {{RoundBoxBottom}} [[Emotion|Emotions]] are a part of what makes us human and affect us all differently (Izard, 2013). Emotions effect the entirety of a person including their body, perception, cognition, their actions, personality and even their relationships (Izard, 2013). Emotions alert us to sensory information that needs our attention to keep us safe and healthy mentally and physically and steers our actions in these times. Although emotions are very helpful and necessary, they can also have negative impact when they become maladaptive or last too long. This is when we need to regulate our emotions. Emotion regulation is a helpful and important process for humans to function effectively day to day (''Emotion regulation: Conceptual foundations'', 2007). [[File:Plutchik-wheel.svg|thumb|'''Figure 2'''. Plutchik's wheel of emotions {{ic|Explain relevance to the topic}}]] Emotions can be negative or positive to a person and can be triggered by a range of situations in our lives, some we can't control, but some we can. So what we do to keep our emotions from becoming too much? {{RoundBoxTop|theme=3}}'''Focus questions''' * What is emotion regulation? * How can exercise help regulate emotions? * What exercise is best to help regulate emotions? {{RoundBoxBottom}} == What is emotion regulation? == [https://www.paulekman.com/universal-emotions/ Universal emotions] were identified by Paul Ekman, that we ''all'' feel, despite we speak different languages, live in different place regional, have different cultural, and ethnic differences (Ekman et al., 1999). He included happiness/enjoyment, anger, fear, disgust, sadness, surprise and then later added contempt. Plutchik went on to create a wheel of emotions to help visually explain these different emotions and show further breakdown ("The emotions," 1991). Emotions can also become dysregulated. [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional dysregulation] is when your emotions feel too big and too strong and a person loses the ability to control their behaviour. Emotional dysregulation is the inability to be aware of, accept, regulate, and modify emotional reactions and subsequent behaviours (Gratz & Roemer, 2004). It can look and feel like irritability, anger, mood swings, outbursts, feeling numb or detached, shutting down. Or in more extreme cases aggressive behaviors, stained relationships and connections and verbals shouting and crying<ref>{{Cite web|url=https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation|title=Cleveland Clinic: Emotional-dysregulation}}</ref>. The goal of emotion regulation is often explained as down-regulating (reducing) the sensation and/or duration of negative emotions such as feeling sad, angry or anxious (Gross et al., 2006). People can also try to up-regulate and increase the intensity or duration of positive emotions (Quoidbach et al., 2010). Emotion regulation is a very important skill of which children typically struggle with and some this can follow into adolescents and adulthood for some. == How can exercise help to regulate emotions? == Exercise is an essential part of humans everyday lives. It's recommended by [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians The Department of Health, Disability and Aging to move most days].  {{RoundBoxTop|theme=2}}'''Department of Health, Disability and Aging recommended physical activity for adults''' -Moderate to vigorous physical activities for at least 30 minutes on most days. -Muscle-strengthening activities at least twice per week. -Functional activities targeting mobility, balance, and coordination on 3 or more days per week. -Several hours of light-intensity physical activity daily. {{RoundBoxBottom}} === Physiological effects === Exercise positively attributes to well emotional regulation by changing brain chemistry, strengthening brain networks, and reducing stress hormones. Exercise has been proven to reduce cortisol levels and increase dopamine and other endorphins which are hormones responsible for the feelings of calm and comfort. Simply said, exercise puts the brain in a state where it is better able to regulate emotions! (reference) * '''Reduces cortisol and increases dopamine''': Meaning emotional regulation is much easier. (reference) * '''Improve sleep:''' Exercise promotes better sleep as physical exertion regulates stress, lowers stress hormones and reduces muscle tension. === Psychological effects === Psychological benefits of exercise have continued to be support in many ways such as: * '''Mindfulness:''' Exercise can give people oppotunities to be present in their movements and have a healthy break from negative of challenging mental strains. * '''Connection:''' Exercise can be a fantastic way to connect with others and creat social support networks. Walking with friends or family is a perfect way to end a tough day. * '''Building self-efficacy and self-belief:''' Setting and reaching fitness goals can build personal xonfidence, build resilitience and give you a sense of control over your daily structure. Building self-efficacy and self-esteem through exercise is an important and healthy way to cope with daily challenges. * '''Improved mental health:''' Exercise has been proven to improve mental health. == Psychological theories == Many psychological theories can be applied to exercise and emotional regulation however here we will cover the process model of emotion regulation, [[w:Self determination theory|self-determination theory]] (SDT), [[cognitive evaluation theory]] (CET), [[drive theory|drive theory/ Conservation of Resources (COR) Theory]] and xxxxxxx. === Process model of emotion regulation (Gross, 19XX) === Gross's (19XX) Process model of emotion regulation in one of the most influential frameworks for understanding how emotions arise and how we can regulate them. === Self-determination theory (Deci & Ryan, 1985) === The self-determination theory focuses on fulfilling three core psychological needs, autonomy, competency and relatedness (need reference??). The SDT can be applied to exercise thanks to the metadata review by Teixeira et al., 2012. # <u>Autonomy:</u> Studies have consistently supported SDT aspects showing a positive relationship between more autonomous types of motivation and exercise. Autonomous motivation in terms of exercise, means excercising because you ''want'' to, not because you have to. Autonmous motvation towards exercise is a key mindset for consistency to develop. Individuals need to feel in control of their own actions, choices and so should choose their own exercise that suits them personally. The individual should be in charge of implementing structures and routines that work for them. # <u>Competence:</u> Competence satisfaction predicts better exercise participation (Teixeira et al., 2012). People are more likely to exercise if they experience competence satisfaction and feel like they are good at what they are doing. Individuals need to feel competent, can master challenging situations and feel as though they are improving and feel able to reach their own goals. This can be applied to fitness goals. # <u>Relatedness:</u> Relatedness is the need to experience a sense of belonging, connection, and care for others. People want to feel accepted, included and a part of a group. Relatedness is the desire to feel connected and interpersonally involved in warm, caring, reciprocal relationships. Exercise is a fantastic way to not just get the physical benefit of moving your body, but also the social and connection of doing so, alongside others with a common health goal. Joining a run club, aqua aerobics or tai chi class can help make social connections whilst moving your body. ==== Cognitive evaluation theory (Deci & Ryan, 1985) ==== Cognitive evaluation theory is a sub-theory of self-determination theory that explains how external events influence intrinsic motivation by affecting individual <u>perceived</u> competence and autonomy. Edward L. Deci and Richard M. Ryan develop CET in 1985, with the focus on the interplay between intrinsic and extrinsic motivation. In short, what drives human motiovation. CET examines how external factors—such as rewards, feedback, deadlines, or social pressures—can either enhance or undermine a person’s natural drive to engage in an activity for its own sake. The theory emphasizes that intrinsic motivation is not fixed but highly sensitive to environmental cues. In an sports or exercise context, individuals or coaches can use CET principles to encourage skill development and enjoyment rather than relying solely on external rewards such as first place trophies. Types of motivation from cognitive evaluation theory: * '''Intrinsic motivation:''' Doing an activity purely for its satisfaction, enjoyment, or interest predicts better exercise participation of long-term exercise adherence. A trend towards identified regulation (exercising because you see value in doing so) was found to predict participation more strongly than intrinsic motivation in the short term. * '''Extrinsic motivation:''' Doing an activity for external rewards, pressures, or avoidance of consequences, which can range from controlled (forced) to autonomous (internalised values). This is a good way to start exercising however intrinsic motives are better for long-term exercise habits. Extrinsic motives include winning a medal, gaining respect and praise or losing weight. * '''Amotivation:''' Amotivation is complete lack of intent or drive to take action. Three aspects attribute to feeling amotivative: Low competence, low autonomy, and low relatedness. The amotivated state is a predictor of poor performance and poor functioning. === Drive theory (Hull, 1943) === The drive theory proposes that humans are motivated to reduces pressures and stress and discomfort. Hull (1943) suggests that as a individuals arousal level increase, performance will also increase. Alternatively, if an individual's arousal levels decrease, then the performance would also decrease. Although this is not a primary motivational theory in the modern time, it still gives a good overview on what modern individuals often do not choose to exercise in time of emotional dysregulation. People are unconsciously looking to preserve energy and reduce stresses in the short term. What makes drive theory a learning theory, not just a motivation theory, is the feedback loop. When a particular behavior successfully reduces a drive, the connection between the situation and that behavior gets stronger. Hull called this “habit strength.” This is another point that is related for modern day people, we need to consciously turn to strengthen healthy habits and weaken maladaptive automatic responses.{{RoundBoxTop|theme=3}}'''So how can the self-determination theory be applied to Sarah?''' Autonomy: Sarah hates running, but she loves dancing. Zumba classes at 7:30pm twice a week might suit Sarah better after she has a quick tea and a rest straight after work. Competency: Sarah hasn't even done dance classes before, but she does feel confident in showing off her moves. Zumba classes could motivate her to move her body while building more range in her dance move repertoire. Relatedness: Sarah is a very social person, and some of her friends might even like to join the Zumba classes too. Intrinsic motivators: Sarah should focus on aerobic exercises that are enjoyable to her. Looking for external motivators like weight loss or looking a sort of way, will not help Sarah stick to her movement long-term.{{RoundBoxBottom}} # == What exercise is best to help regulate emotions? == Australian adults should be active daily, undertaking a variety of physical activities at different intensities, which can include<ref>{{Cite web|url=https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians/recommendations-for-adults-18-to-64-years?language=en|title=Physical Activity Recommendations for Australian adults|date=|last=Australian Government}}</ref>: * moderate- to vigorous-intensity physical activities for 30 minutes or more on most days * muscle-strengthening activities on 2 or more days per week * functional activities targeting mobility, balance, and coordination on 3 or more days per week * several hours of light-intensity physical activity daily. Any movement is beneficial to individuals but to regulate your emotions the most ideal exercise looks like this: Anaerobic exercise (also known as resistance training) has benefits on emotion regulation such as....... Aerobic exercise<ref>{{Cite web|url=https://my.clevelandclinic.org/health/articles/7050-aerobic-exercise|title=Aerobic Exercise|last=Cleveland Clinic}}</ref> is when you continuously use oxygen as your fuel. Aerobic exercise promotes emotion regulation (Wang et al., 2024). And what is more important to note is that there are <u>immediate</u> results on mood. * {| class="wikitable" |+'''Table 1:''' Optimal exercise for each age category, with supporting studies. !Age !Exercise !Intensity/ Frequency !Study |- |Children |A variety of exercise such as running, soccer, dancing, tips, games and anything else that sustains interest and fun of the child. |60 minutes daily minimum Moderate–vigorous |Chaput et al., 2020 |- |Adolesence |A range of aerobic and strength exercise which can include sports, gym, HIT sessions, weights or running. |60 minutes daily Moderate–vigorous |Salvan et al., 2021 Parrish et al., 2020 |- |Adults |Walking, riding, swimming, running, zumba or sports. Begin with short, low-intensity walking and gradually increase duration and intensity |30 minutes or more, most days Moderate–vigorous strength twice a week |Ligeza et al., 2023, Mizzi et al., 2022 |- |Older Adults |Walking, exercise bike or water aerobics |Set depending on exercise ability and history |Erickson et al., 2011 |} ;Test yourself<quiz display="simple"> {High intensity running is best for older adults to help manager their emotions: |type="()"} + True - False {Children should take part in more frequent exercise than adults and it should be more varied: |type="()"} - True + False </quiz> ** ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== In conclusion, physical activity has physical, and psychological benefits and is beneficial in times of emotional dysregulation. Using exercise as a preventative mechanism is also important. '''Useful takeaways:''' * One session of aerobic exercise can significantly boost positive mood and alleviate stress responses. It is recommended for use as an immediate mood regulation strategy. * Any movement can be used such as walking, dancing or sport. *Recommended length: 150 to 330 words {{tip|Key messages * Start by moving your body for 30 minutes in any way you enjoy. Positive effects on mood starts immediately. What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Erickson, K. I., Voss, M. W., Prakash, R. S., Basak, C., Szabo, A., Chaddock, L., Kim, J. S., Heo, S., Alves, H., White, S. M., Wojcicki, T. R., Mailey, E., Vieira, V. J., Martin, S. A., Pence, B. D., Woods, J. A., McAuley, E., & Kramer, A. F. (2011). Exercise training increases size of hippocampus and improves memory. Proceedings of the National Academy of Sciences, 108(7), 3017-3022. https://doi.org/10.1073/pnas.1015950108 Gratz, K. L., & Roemer, L. (2004). Multidimensional Assessment of Emotion Regulation and Dysregulation: Development, Factor Structure, and Initial Validation of the Difficulties in Emotion Regulation Scale. Journal of Psychopathology and Behavioral Assessment, 26(1), 41-54. https://doi.org/10.1023/B:JOBA.0000007455.08539.94 Izard, C. E. (2013). Human emotions. Springer Science & Business Media. Ligeza, T. S., Maciejczyk, M., Wyczesany, M., & Junghofer, M. (2023). The effects of a single aerobic exercise session on mood and neural emotional reactivity in depressed and healthy young adults: A late positive potential study. Psychophysiology, 60(1), e14137. https://doi.org/https://doi.org/10.1111/psyp.14137 Salvan, P., Wassenaar, T., Wheatley, C., Beale, N., Cottaar, M., Papp, D., Bastiani, M., Fitzgibbon, S., Duff, E., Andersson, J., Winkler, A. M., Douaud, G., Nichols, T. E., Smith, S., Dawes, H., & Johansen-Berg, H. (2021). Multimodal Imaging Brain Markers in Early Adolescence Are Linked with a Physically Active Lifestyle. The Journal of Neuroscience, 41(5), 1092. https://doi.org/10.1523/JNEUROSCI.1260-20.2020 Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., & Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: a systematic review. International journal of behavioral nutrition and physical activity, 9(1), 78. Wang, X., Liu, T., Jin, X., & Zhou, C. (2024). Aerobic exercise promotes emotion regulation: a narrative review. Experimental Brain Research, 242(4), 783-796. https://doi.org/https://doi.org/10.1007/s00221-024-06791-1 }} ==External links== External relevant resources: * [https://www.paulekman.com/universal-emotions/ Universal Emotions | What are Emotions? | Paul Ekman Group] * [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional Dysregulation: What It Is, Causes & Treatment] * [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians 24-hour movement guidelines for all Australians | Australian Government Department of Health, Disability and Ageing] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] ==See also== * [[Motivation and emotion/Book/2025/Emotion regulation through exercise|Emotion regulation through exercise]] (Book chapter, 2025) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional self-regulation]] [[Category:Motivation and emotion/Book/Exercise]] <noinclude> <hr> __TOC__ [[Category:Motivation and emotion/Book]] cyu9v71379tfoesf9eq18qnlem7jfyx 2835030 2835027 2026-09-29T10:52:00Z KB3250298 3105557 /* References */ 2835030 wikitext text/x-wiki {{title|Emotion regulation through exercise:<br>How do people use exercise to regulate their emotional states?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}}'''Case study''' [[File:Laptop and girl biting pencil-pixabay.jpg|thumb|200px|'''Figure 1'''. Sarah feeling very unregulated with too much going on in her life.]] Sarah stayed up late last night studying as she has three assignments due in the same week. When her alarm goes off at 6am for work she feels exhausted so crawls to the shower to wake herself up. Sarahs house mate bangs on the door yelling to hurry up as they need to use the bathroom. Sarah is already sick of this day but gets dressed and rushes off to her part-time work at a before school care center. Her boss grumbles at her for being 10 minutes late and the children are complaining and fighting about the breakfast they served today. Sarah pushes though her shift, and when it's over, she gets back to her car and sits in it for a moment, She feels tired, grumpy and so frustrated with so many people right now she feels like she might explode. Sarah knows she should go for a run because she hasn't exercised all week, but she is just too tired. Maybe she will get Macca's on the way home and go to her couch to scroll on her phone while she is peacefully at home by herself. What should Sarah do to help regulate her rising emotions: Go for a run or home to scroll? {{RoundBoxBottom}} [[Emotion|Emotions]] are a part of what makes us human and affect us all differently (Izard, 2013). Emotions effect the entirety of a person including their body, perception, cognition, their actions, personality and even their relationships (Izard, 2013). Emotions alert us to sensory information that needs our attention to keep us safe and healthy mentally and physically and steers our actions in these times. Although emotions are very helpful and necessary, they can also have negative impact when they become maladaptive or last too long. This is when we need to regulate our emotions. Emotion regulation is a helpful and important process for humans to function effectively day to day (''Emotion regulation: Conceptual foundations'', 2007). [[File:Plutchik-wheel.svg|thumb|'''Figure 2'''. Plutchik's wheel of emotions {{ic|Explain relevance to the topic}}]] Emotions can be negative or positive to a person and can be triggered by a range of situations in our lives, some we can't control, but some we can. So what we do to keep our emotions from becoming too much? {{RoundBoxTop|theme=3}}'''Focus questions''' * What is emotion regulation? * How can exercise help regulate emotions? * What exercise is best to help regulate emotions? {{RoundBoxBottom}} == What is emotion regulation? == [https://www.paulekman.com/universal-emotions/ Universal emotions] were identified by Paul Ekman, that we ''all'' feel, despite we speak different languages, live in different place regional, have different cultural, and ethnic differences (Ekman et al., 1999). He included happiness/enjoyment, anger, fear, disgust, sadness, surprise and then later added contempt. Plutchik went on to create a wheel of emotions to help visually explain these different emotions and show further breakdown ("The emotions," 1991). Emotions can also become dysregulated. [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional dysregulation] is when your emotions feel too big and too strong and a person loses the ability to control their behaviour. Emotional dysregulation is the inability to be aware of, accept, regulate, and modify emotional reactions and subsequent behaviours (Gratz & Roemer, 2004). It can look and feel like irritability, anger, mood swings, outbursts, feeling numb or detached, shutting down. Or in more extreme cases aggressive behaviors, stained relationships and connections and verbals shouting and crying<ref>{{Cite web|url=https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation|title=Cleveland Clinic: Emotional-dysregulation}}</ref>. The goal of emotion regulation is often explained as down-regulating (reducing) the sensation and/or duration of negative emotions such as feeling sad, angry or anxious (Gross et al., 2006). People can also try to up-regulate and increase the intensity or duration of positive emotions (Quoidbach et al., 2010). Emotion regulation is a very important skill of which children typically struggle with and some this can follow into adolescents and adulthood for some. == How can exercise help to regulate emotions? == Exercise is an essential part of humans everyday lives. It's recommended by [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians The Department of Health, Disability and Aging to move most days].  {{RoundBoxTop|theme=2}}'''Department of Health, Disability and Aging recommended physical activity for adults''' -Moderate to vigorous physical activities for at least 30 minutes on most days. -Muscle-strengthening activities at least twice per week. -Functional activities targeting mobility, balance, and coordination on 3 or more days per week. -Several hours of light-intensity physical activity daily. {{RoundBoxBottom}} === Physiological effects === Exercise positively attributes to well emotional regulation by changing brain chemistry, strengthening brain networks, and reducing stress hormones. Exercise has been proven to reduce cortisol levels and increase dopamine and other endorphins which are hormones responsible for the feelings of calm and comfort. Simply said, exercise puts the brain in a state where it is better able to regulate emotions! (reference) * '''Reduces cortisol and increases dopamine''': Meaning emotional regulation is much easier. (reference) * '''Improve sleep:''' Exercise promotes better sleep as physical exertion regulates stress, lowers stress hormones and reduces muscle tension. === Psychological effects === Psychological benefits of exercise have continued to be support in many ways such as: * '''Mindfulness:''' Exercise can give people oppotunities to be present in their movements and have a healthy break from negative of challenging mental strains. * '''Connection:''' Exercise can be a fantastic way to connect with others and creat social support networks. Walking with friends or family is a perfect way to end a tough day. * '''Building self-efficacy and self-belief:''' Setting and reaching fitness goals can build personal xonfidence, build resilitience and give you a sense of control over your daily structure. Building self-efficacy and self-esteem through exercise is an important and healthy way to cope with daily challenges. * '''Improved mental health:''' Exercise has been proven to improve mental health. == Psychological theories == Many psychological theories can be applied to exercise and emotional regulation however here we will cover the process model of emotion regulation, [[w:Self determination theory|self-determination theory]] (SDT), [[cognitive evaluation theory]] (CET), [[drive theory|drive theory/ Conservation of Resources (COR) Theory]] and xxxxxxx. === Process model of emotion regulation (Gross, 19XX) === Gross's (19XX) Process model of emotion regulation in one of the most influential frameworks for understanding how emotions arise and how we can regulate them. === Self-determination theory (Deci & Ryan, 1985) === The self-determination theory focuses on fulfilling three core psychological needs, autonomy, competency and relatedness (need reference??). The SDT can be applied to exercise thanks to the metadata review by Teixeira et al., 2012. # <u>Autonomy:</u> Studies have consistently supported SDT aspects showing a positive relationship between more autonomous types of motivation and exercise. Autonomous motivation in terms of exercise, means excercising because you ''want'' to, not because you have to. Autonmous motvation towards exercise is a key mindset for consistency to develop. Individuals need to feel in control of their own actions, choices and so should choose their own exercise that suits them personally. The individual should be in charge of implementing structures and routines that work for them. # <u>Competence:</u> Competence satisfaction predicts better exercise participation (Teixeira et al., 2012). People are more likely to exercise if they experience competence satisfaction and feel like they are good at what they are doing. Individuals need to feel competent, can master challenging situations and feel as though they are improving and feel able to reach their own goals. This can be applied to fitness goals. # <u>Relatedness:</u> Relatedness is the need to experience a sense of belonging, connection, and care for others. People want to feel accepted, included and a part of a group. Relatedness is the desire to feel connected and interpersonally involved in warm, caring, reciprocal relationships. Exercise is a fantastic way to not just get the physical benefit of moving your body, but also the social and connection of doing so, alongside others with a common health goal. Joining a run club, aqua aerobics or tai chi class can help make social connections whilst moving your body. ==== Cognitive evaluation theory (Deci & Ryan, 1985) ==== Cognitive evaluation theory is a sub-theory of self-determination theory that explains how external events influence intrinsic motivation by affecting individual <u>perceived</u> competence and autonomy. Edward L. Deci and Richard M. Ryan develop CET in 1985, with the focus on the interplay between intrinsic and extrinsic motivation. In short, what drives human motiovation. CET examines how external factors—such as rewards, feedback, deadlines, or social pressures—can either enhance or undermine a person’s natural drive to engage in an activity for its own sake. The theory emphasizes that intrinsic motivation is not fixed but highly sensitive to environmental cues. In an sports or exercise context, individuals or coaches can use CET principles to encourage skill development and enjoyment rather than relying solely on external rewards such as first place trophies. Types of motivation from cognitive evaluation theory: * '''Intrinsic motivation:''' Doing an activity purely for its satisfaction, enjoyment, or interest predicts better exercise participation of long-term exercise adherence. A trend towards identified regulation (exercising because you see value in doing so) was found to predict participation more strongly than intrinsic motivation in the short term. * '''Extrinsic motivation:''' Doing an activity for external rewards, pressures, or avoidance of consequences, which can range from controlled (forced) to autonomous (internalised values). This is a good way to start exercising however intrinsic motives are better for long-term exercise habits. Extrinsic motives include winning a medal, gaining respect and praise or losing weight. * '''Amotivation:''' Amotivation is complete lack of intent or drive to take action. Three aspects attribute to feeling amotivative: Low competence, low autonomy, and low relatedness. The amotivated state is a predictor of poor performance and poor functioning. === Drive theory (Hull, 1943) === The drive theory proposes that humans are motivated to reduces pressures and stress and discomfort. Hull (1943) suggests that as a individuals arousal level increase, performance will also increase. Alternatively, if an individual's arousal levels decrease, then the performance would also decrease. Although this is not a primary motivational theory in the modern time, it still gives a good overview on what modern individuals often do not choose to exercise in time of emotional dysregulation. People are unconsciously looking to preserve energy and reduce stresses in the short term. What makes drive theory a learning theory, not just a motivation theory, is the feedback loop. When a particular behavior successfully reduces a drive, the connection between the situation and that behavior gets stronger. Hull called this “habit strength.” This is another point that is related for modern day people, we need to consciously turn to strengthen healthy habits and weaken maladaptive automatic responses.{{RoundBoxTop|theme=3}}'''So how can the self-determination theory be applied to Sarah?''' Autonomy: Sarah hates running, but she loves dancing. Zumba classes at 7:30pm twice a week might suit Sarah better after she has a quick tea and a rest straight after work. Competency: Sarah hasn't even done dance classes before, but she does feel confident in showing off her moves. Zumba classes could motivate her to move her body while building more range in her dance move repertoire. Relatedness: Sarah is a very social person, and some of her friends might even like to join the Zumba classes too. Intrinsic motivators: Sarah should focus on aerobic exercises that are enjoyable to her. Looking for external motivators like weight loss or looking a sort of way, will not help Sarah stick to her movement long-term.{{RoundBoxBottom}} # == What exercise is best to help regulate emotions? == Australian adults should be active daily, undertaking a variety of physical activities at different intensities, which can include<ref>{{Cite web|url=https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians/recommendations-for-adults-18-to-64-years?language=en|title=Physical Activity Recommendations for Australian adults|date=|last=Australian Government}}</ref>: * moderate- to vigorous-intensity physical activities for 30 minutes or more on most days * muscle-strengthening activities on 2 or more days per week * functional activities targeting mobility, balance, and coordination on 3 or more days per week * several hours of light-intensity physical activity daily. Any movement is beneficial to individuals but to regulate your emotions the most ideal exercise looks like this: Anaerobic exercise (also known as resistance training) has benefits on emotion regulation such as....... Aerobic exercise<ref>{{Cite web|url=https://my.clevelandclinic.org/health/articles/7050-aerobic-exercise|title=Aerobic Exercise|last=Cleveland Clinic}}</ref> is when you continuously use oxygen as your fuel. Aerobic exercise promotes emotion regulation (Wang et al., 2024). And what is more important to note is that there are <u>immediate</u> results on mood. * {| class="wikitable" |+'''Table 1:''' Optimal exercise for each age category, with supporting studies. !Age !Exercise !Intensity/ Frequency !Study |- |Children |A variety of exercise such as running, soccer, dancing, tips, games and anything else that sustains interest and fun of the child. |60 minutes daily minimum Moderate–vigorous |Chaput et al., 2020 |- |Adolesence |A range of aerobic and strength exercise which can include sports, gym, HIT sessions, weights or running. |60 minutes daily Moderate–vigorous |Salvan et al., 2021 Parrish et al., 2020 |- |Adults |Walking, riding, swimming, running, zumba or sports. Begin with short, low-intensity walking and gradually increase duration and intensity |30 minutes or more, most days Moderate–vigorous strength twice a week |Ligeza et al., 2023, Mizzi et al., 2022 |- |Older Adults |Walking, exercise bike or water aerobics |Set depending on exercise ability and history |Erickson et al., 2011 |} ;Test yourself<quiz display="simple"> {High intensity running is best for older adults to help manager their emotions: |type="()"} + True - False {Children should take part in more frequent exercise than adults and it should be more varied: |type="()"} - True + False </quiz> ** ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== In conclusion, physical activity has physical, and psychological benefits and is beneficial in times of emotional dysregulation. Using exercise as a preventative mechanism is also important. '''Useful takeaways:''' * One session of aerobic exercise can significantly boost positive mood and alleviate stress responses. It is recommended for use as an immediate mood regulation strategy. * Any movement can be used such as walking, dancing or sport. *Recommended length: 150 to 330 words {{tip|Key messages * Start by moving your body for 30 minutes in any way you enjoy. Positive effects on mood starts immediately. What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Chaput, J. P., Willumsen, J., Bull, F., Chou, R., Ekelund, U., Firth, J., Jago, R., Ortega, F. B., & Katzmarzyk, P. T. (2020). 2020 WHO guidelines on physical activity and sedentary behaviour for children and adolescents aged 5-17 years: summary of the evidence. Int J Behav Nutr Phys Act, 17(1), 141. https://doi.org/10.1186/s12966-020-01037-z Erickson, K. I., Voss, M. W., Prakash, R. S., Basak, C., Szabo, A., Chaddock, L., Kim, J. S., Heo, S., Alves, H., White, S. M., Wojcicki, T. R., Mailey, E., Vieira, V. J., Martin, S. A., Pence, B. D., Woods, J. A., McAuley, E., & Kramer, A. F. (2011). Exercise training increases size of hippocampus and improves memory. Proceedings of the National Academy of Sciences, 108(7), 3017-3022. https://doi.org/10.1073/pnas.1015950108 Gratz, K. L., & Roemer, L. (2004). Multidimensional Assessment of Emotion Regulation and Dysregulation: Development, Factor Structure, and Initial Validation of the Difficulties in Emotion Regulation Scale. Journal of Psychopathology and Behavioral Assessment, 26(1), 41-54. https://doi.org/10.1023/B:JOBA.0000007455.08539.94 Izard, C. E. (2013). Human emotions. Springer Science & Business Media. Ligeza, T. S., Maciejczyk, M., Wyczesany, M., & Junghofer, M. (2023). The effects of a single aerobic exercise session on mood and neural emotional reactivity in depressed and healthy young adults: A late positive potential study. Psychophysiology, 60(1), e14137. https://doi.org/https://doi.org/10.1111/psyp.14137 Mizzi, A. L., McKinnon, M. C., & Becker, S. (2022). The Impact of Aerobic Exercise on Mood Symptoms in Trauma-Exposed Young Adults: A Pilot Study [Original Research]. Frontiers in Behavioral Neuroscience, Volume 16 - 2022. https://doi.org/10.3389/fnbeh.2022.829571 Parrish, A.-M., Tremblay, M. S., Carson, S., Veldman, S. L., Cliff, D., Vella, S., Chong, K. H., Nacher, M., del Pozo Cruz, B., & Ellis, Y. (2020). Comparing and assessing physical activity guidelines for children and adolescents: a systematic literature review and analysis. International journal of behavioral nutrition and physical activity, 17(1), 16. Salvan, P., Wassenaar, T., Wheatley, C., Beale, N., Cottaar, M., Papp, D., Bastiani, M., Fitzgibbon, S., Duff, E., Andersson, J., Winkler, A. M., Douaud, G., Nichols, T. E., Smith, S., Dawes, H., & Johansen-Berg, H. (2021). Multimodal Imaging Brain Markers in Early Adolescence Are Linked with a Physically Active Lifestyle. The Journal of Neuroscience, 41(5), 1092. https://doi.org/10.1523/JNEUROSCI.1260-20.2020 Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., & Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: a systematic review. International journal of behavioral nutrition and physical activity, 9(1), 78. Wang, X., Liu, T., Jin, X., & Zhou, C. (2024). Aerobic exercise promotes emotion regulation: a narrative review. Experimental Brain Research, 242(4), 783-796. https://doi.org/https://doi.org/10.1007/s00221-024-06791-1 }} == External links == External relevant resources: * [https://www.paulekman.com/universal-emotions/ Universal Emotions | What are Emotions? | Paul Ekman Group] * [https://my.clevelandclinic.org/health/symptoms/25065-emotional-dysregulation Emotional Dysregulation: What It Is, Causes & Treatment] * [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians 24-hour movement guidelines for all Australians | Australian Government Department of Health, Disability and Ageing] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] ==See also== * [[Motivation and emotion/Book/2025/Emotion regulation through exercise|Emotion regulation through exercise]] (Book chapter, 2025) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional self-regulation]] [[Category:Motivation and emotion/Book/Exercise]] <noinclude> <hr> __TOC__ [[Category:Motivation and emotion/Book]] iznfdub4vuhzc4vi1ngxr5w4jb1sywa 10 330233 2834957 2833740 2026-09-29T03:41:21Z Jtneill 10242 2834957 wikitext text/x-wiki <noinclude>{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].<br><br>[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}} __NOTOC__</noinclude><includeonly> ==Topic development feedback== {{RoundBoxTop|theme=8}} The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date. {{RoundBoxBottom}} {{RoundBoxTop|theme=9}} [[File:Autoroute icone.svg|right|85px]] ===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]=== {{{1|No comment}}} ===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]=== {{{2|No comment}}} ===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]=== {{{3|No comment}}} ===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]=== {{{4|No comment}}} ===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]=== {{{5|No comment}}} ===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]=== {{{6|No comment}}} ===7. [[Motivation and emotion/Assessment/Topic#References|References]]=== {{{7|No comment}}} ===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]=== {{{8|No comment}}} ===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]=== {{{9|No comment}}} ===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]=== {{{10|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> # |3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} ~~~~ </pre> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> #|3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:42, 17 August 2025 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template which includes commonly used feedback comments: <small><small><small><pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated[[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying <!-- Conceptual --> # Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Make the relevance of the scenario to the topic more clear # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} ~~~~ </pre></small></small></small> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying <!-- Conceptual --> # Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:58, 26 August 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Template:MEBF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Topic]] </noinclude> mnkaxr2hap2m1rd3u529db5qbk532fz 2834958 2834957 2026-09-29T03:44:51Z Jtneill 10242 2834958 wikitext text/x-wiki <noinclude>{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].<br><br>[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}} __NOTOC__</noinclude><includeonly> ==Topic development feedback== {{RoundBoxTop|theme=8}} The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date. {{RoundBoxBottom}} {{RoundBoxTop|theme=9}} [[File:Autoroute icone.svg|right|85px]] ===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]=== {{{1|No comment}}} ===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]=== {{{2|No comment}}} ===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]=== {{{3|No comment}}} ===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]=== {{{4|No comment}}} ===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]=== {{{5|No comment}}} ===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]=== {{{6|No comment}}} ===7. [[Motivation and emotion/Assessment/Topic#References|References]]=== {{{7|No comment}}} ===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]=== {{{8|No comment}}} ===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]=== {{{9|No comment}}} ===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]=== {{{10|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> # |3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} ~~~~ </pre> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> #|3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:42, 17 August 2025 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template which includes commonly used feedback comments: <small><small><small><pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Conceptual --> # Messy [[Motivation and emotion/Assessment/Major project/Structure|heading structure]] – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Make the relevance of the scenario to the topic more clear # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} ~~~~ </pre></small></small></small> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Conceptual --> # Messy [[Motivation and emotion/Assessment/Major project/Structure|heading structure]] – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:58, 26 August 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Template:MEBF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Topic]] </noinclude> r22ps071wbq17gl3mqaim8ambv42qqm 2834959 2834958 2026-09-29T03:47:19Z Jtneill 10242 2834959 wikitext text/x-wiki <noinclude>{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].<br><br>[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}} __NOTOC__</noinclude><includeonly> ==Topic development feedback== {{RoundBoxTop|theme=8}} The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date. {{RoundBoxBottom}} {{RoundBoxTop|theme=9}} [[File:Autoroute icone.svg|right|85px]] ===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]=== {{{1|No comment}}} ===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]=== {{{2|No comment}}} ===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]=== {{{3|No comment}}} ===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]=== {{{4|No comment}}} ===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]=== {{{5|No comment}}} ===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]=== {{{6|No comment}}} ===7. [[Motivation and emotion/Assessment/Topic#References|References]]=== {{{7|No comment}}} ===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]=== {{{8|No comment}}} ===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]=== {{{9|No comment}}} ===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]=== {{{10|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> # |3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} ~~~~ </pre> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> #|3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:42, 17 August 2025 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template which includes commonly used feedback comments: <small><small><small><pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Conceptual --> # Messy [[Motivation and emotion/Assessment/Major project/Structure|heading structure]] – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Make the relevance of the scenario to the topic more clear # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} ~~~~ </pre></small></small></small> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Conceptual --> # Messy [[Motivation and emotion/Assessment/Major project/Structure|heading structure]] – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:58, 26 August 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Template:MEBF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Topic]] </noinclude> 3jwjfjqfuwbtz0r88p4tl7chklpcofo 2834961 2834959 2026-09-29T03:53:14Z Jtneill 10242 # If these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]], this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. 2834961 wikitext text/x-wiki <noinclude>{{note|Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise for [[motivation and emotion]].<br><br>[[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]].}} __NOTOC__</noinclude><includeonly> ==Topic development feedback== {{RoundBoxTop|theme=8}} The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date. {{RoundBoxBottom}} {{RoundBoxTop|theme=9}} [[File:Autoroute icone.svg|right|85px]] ===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]=== {{{1|No comment}}} ===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]=== {{{2|No comment}}} ===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]=== {{{3|No comment}}} ===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]=== {{{4|No comment}}} ===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]=== {{{5|No comment}}} ===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]=== {{{6|No comment}}} ===7. [[Motivation and emotion/Assessment/Topic#References|References]]=== {{{7|No comment}}} ===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]=== {{{8|No comment}}} ===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]=== {{{9|No comment}}} ===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]=== {{{10|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> # |3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} ~~~~ </pre> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> #|3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:42, 17 August 2025 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template which includes commonly used feedback comments: <small><small><small><pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Conceptual --> # Messy [[Motivation and emotion/Assessment/Major project/Structure|heading structure]] – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. # If these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]], this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Make the relevance of the scenario to the topic more clear # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} ~~~~ </pre></small></small></small> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or subtitle missing, not correctly worded, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) <!-- 3-level --> # Overly complicated [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Conceptual --> # Messy [[Motivation and emotion/Assessment/Major project/Structure|heading structure]] – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # The headings lack sufficient incision into, and exposition of, the topic # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Research is not needed as a separate section; instead, integrate and synthesise relevant research throughout the chapter alongside the theoretical perspectives and real-life applications <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # Check grammar (e.g,. missing question mark) <!-- Remove --> # Remove sub-headings in the Overview and/or Conclusion # Remove "Introduction" heading – provide this information in Overview or move into subsequent sections # Remove definition(s) headings; cover this in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Remove case study/scenario headings; instead embed the case study/scenario(s) within relevant section(s) # Remove quiz headings; instead embed quiz question(s) within relevant section(s) # Remove numbers from headings # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Reasonably good # Basic # Insufficient # Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. # If these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]], this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # An engaging scenario with a relevant image is planned or presented in a feature box at start of the Overview # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section # Promising scenario; would benefit from further development to make it more compelling and illustrative # The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population # I moved an image into the feature box to help attract reader interest # Choose an image that illustrates the scenario. Move theoretical diagrams into a subsequent section. # Add an image to the scenario to help attract reader interest # Put the scenario or case study into a feature box at the start of this section (fixed) # Make the scenario concrete and down-to-earth, using a realistic real-world example of the target psychological phenomenon rather than an abstract description. # Make the relevance of the scenario to the topic more clear. It should be obvious to a reader how the scenario illustrates the topic (i.e., the sub-title question). Revise the scenario to make this connection more explicit. # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # What are the most relevant psychological theories? # Briefly overview the main theoretical perspectives relevant to the topic # Introduce topic using plain English; most citations can be moved into subsequent setions # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person point of view for main body text (except 1st/2nd person point of view can work within feature boxes for scenarios) <!-- Focus questions --> <!-- Alignment and quality --> # The focus questions are well aligned with the sub-title and top-level headings # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Promising focus questions # There is no need for a question dedicated to research; instead, integrate covered of research throughout # There are too many focus questions; only include questions that are needed to address the sub-title; avoid side-quests—these are usually better treated as examples or scenarios, or left out altogether so that the chapter can concentrate on the core task of synthesising the best psychological theory and research about the topic. <!-- Other --> # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) # The writing style is typical of AI-generated material with minimal human oversight. See [[Motivation and emotion/Assessment/Using generative AI|genAI content]] for guidance about ethical and professional practice. <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Balance theoretical content with critical synthesis of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Promising use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting sentences with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using [https://support.studiosity.com/en/articles/489263-writing-feedback-how-it-works Studiosity Writing Feedback+] via UCLearn or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... (fixed) <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}}—this can also be listed on your user page as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages # Placeholder use of quiz question(s) # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Promising use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Include acknowledgement (e.g., citation(s)) for sources of information presented in the table # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic but effective # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Provide an internal link to the book chapter as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:58, 26 August 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Template:MEBF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Topic]] </noinclude> 4q9hihhp74p9sqx2uh096cbnrpcbffw 0 330317 2834955 2818120 2026-09-29T02:42:35Z James500 297601 /* */ Add 2834955 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of languages. World *Keith Brown and Sarah Ogilvie. Concise Encyclopedia of Languages of the World. Elsevier. 2009. [https://books.google.co.uk/books?id=F2SRqDzB50wC&pg=PP1#v=onepage&q&f=false] *Anatole V Lyovin, Brett Kessler and William R Leben. An Introduction to the Languages of the World. 2nd Ed: 2017: [https://books.google.co.uk/books?id=RQGTDQAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Asya Pereltsvaig. Languages of the World: An Introduction. 2012. [https://books.google.co.uk/books?id=8q06xer0vHkC&pg=PP1#v=onepage&q&f=false] *Merritt Ruhlen. A Guide to the World's Languages. Vol 1 (Classification). Stanford University Press. 1987. [https://books.google.co.uk/books?id=WAMbAAAAIAAJ] *Bernard Comrie. The World's Major Languages. 2nd Ed: 2009: [https://books.google.co.uk/books?id=9S0rDwAAQBAJ&pg=PP1#v=onepage&q&f=false]. *George L. Campbell and Gareth King. Compendium of the World's Languages. 3rd Ed: 2013: [https://books.google.co.uk/books?id=DWAqAAAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Martin D Joachim. Languages of the World: Cataloging Issues and Problems. 1993. [https://books.google.co.uk/books?id=6u18PtO0BoQC&pg=PP1#v=onepage&q&f=false] Origin *Roy Harris. Origin Of Language. 1996. [https://books.google.co.uk/books?id=386lU_0oUWoC&pg=PR3#v=onepage&q&f=false] *James R Hurford. Origins of Language: A Slim Guide. 2014. [https://books.google.co.uk/books?id=InTiAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Merritt Ruhlen. The Origin of Language: Tracing the Evolution of the Mother Tongue. 1994. [https://books.google.com/books?id=retrAAAAIAAJ] *Language Origin: A Multidisciplinary Approach. 1992. [https://books.google.co.uk/books?id=z_yPBAAAQBAJ&pg=PA1933#v=onepage&q&f=false] *Jürgen Trabant and Sean Ward (eds). New Essays on the Origin of Language. 2001. [https://books.google.co.uk/books?id=Pt501C6Zv94C&pg=PP1#v=onepage&q&f=false] *Claire Lefebvre, Bernard Comrie and Henri Cohen (eds). New Perspectives on the Origins of Language. Studies in Language Companion series, vol 144. [https://books.google.co.uk/books?id=S64bAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Denis Bouchard. The Nature and Origin of Language. 2013. [https://books.google.co.uk/books?id=4cRoAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Giorgio Fano. The Origins and Nature of Language. Indiana University Press. [https://books.google.com/books?id=fdlrAAAAIAAJ] *Jean Aitchison. The Seeds of Speech: Language Origin and Evolution. 1996. Canto Ed: 2000. [https://books.google.co.uk/books?id=68Y5gUavbzwC&pg=PP1#v=onepage&q&f=false] *Morris Swadesh. The Origin and Diversification of Language. 2006. 2017. [https://books.google.co.uk/books?id=klUPEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Prehistory *Rudolf Botha and Chris Knight (eds). The Prehistory of Language. 2009. [https://books.google.co.uk/books?id=36tLTfV_hLcC&pg=PP1#v=onepage&q&f=false] *G Révész. The Origins and Prehistory of Language. Longmans, Green and Co. [https://books.google.co.uk/books?id=GxRZAAAAMAAJ] History *Tore Janson. The History of Languages: An Introduction. 2012. [https://books.google.co.uk/books?id=pE2N7noPfEoC&pg=PP1#v=onepage&q&f=false] *Tore Janson. Speak: A Short History of Languages. 2002. [https://books.google.co.uk/books?id=mAgGOU2XmCAC&pg=PP1#v=onepage&q&f=false] *Nicholas Ostler. Empires of the Word: A Language History of the World. Preface dated 2004. [https://books.google.co.uk/books?id=Mz2kxr6v2X4C&pg=PP1#v=onepage&q&f=false] *Steven Roger Fischer. History of Language. 1999. [https://books.google.co.uk/books?id=5i1Ql7QQy0kC&pg=PP1#v=onepage&q&f=false] *A S Diamond. The History and Origin of Language. 1959: [https://books.google.co.uk/books?id=mjcGAQAAIAAJ]. Routledge Revivals. [https://books.google.co.uk/books?id=P5jiEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Henry Sweet. The History of Language. 1900. [https://books.google.co.uk/books?id=PC1GGpv7vlsC&pg=PR3#v=onepage&q&f=false] Social history *Peter Burke and Roy Porter (eds). The Social History of Language. 1987. [https://books.google.co.uk/books?id=oyRshxHVV5sC&pg=PP1#v=onepage&q&f=false] Story *Charles Barber. The Story of Language. Pan Books. 1964. [https://books.google.co.uk/books?id=gx0RAQAAIAAJ] *[[w:en:Mario Pei|Mario Pei]]. The Story of Language. 1949. Lippincott. Revised Ed: 1965. [https://books.google.co.uk/books?id=lqEviMzgv7wC]. Review: [https://books.google.co.uk/books?id=aaCvFv11ZJ4C 67] The Literary Guide 82 (May 1952) Classification *April McMahon and Robert McMahon. Language Classification by Numbers. 2005. [https://books.google.co.uk/books?id=CrEUDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *CF and FM Voegelin. Classification and Index of the World's Languages. (Foundations of Linguistics series). Elsevier. New York. 1977. ISBN 0444001557. [https://books.google.co.uk/books?id=2LAuAAAAYAAJ] Extinct *Johannes Friedrich. Extinct Languages. 1957. [https://books.google.co.uk/books?id=SzcDAAAAMAAJ] *K David Harrison. When Languages Die: The Extinction of the World's Languages and the Erosion of Human Knowledge. 2007. [https://books.google.co.uk/books?id=GTfRCwAAQBAJ&pg=PP1#v=onepage&q&f=false] Dead *Coulter H George. How Dead Languages Work. 2020. [https://books.google.co.uk/books?id=xEfWDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Indo-European languages *Mate Kapović (ed). The Indo-European Languages. 2nd Ed: 2017: [https://books.google.co.uk/books?id=8i0lDwAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Anna Giacalone Ramat and Paolo Ramat (eds). The Indo-European Languages. 1998. [https://books.google.co.uk/books?id=vwUMNCYbLL0C&pg=PP1#v=onepage&q&f=false] **La Lingue Indoeuropee. 1993. *Philip Baldi. An Introduction to the Indo-European Languages. 1983. [https://books.google.co.uk/books?id=lq-mkL23oh8C&pg=PP1#v=onepage&q&f=false] *W B Lockwood. A Panorama of Indo-European Languages. 1972. [https://books.google.co.uk/books?id=QTLMEQAAQBAJ&pg=PA1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=xJ0cAQAAIAAJ] Anatolian languages *Donald C Swanson. A Select Bibliography of the Anatolian Languages. 1948. [https://books.google.co.uk/books?id=OR3KP8kCjzUC] Reprinted from Bulletin of the New York Public Library, [https://books.google.co.uk/books?id=ktkaAAAAMAAJ vol 52], nos 5 and 6, May and June 1948, pp 3 to 26. Hittite *Theo van den Hout. The Elements of Hittite. 2011. [https://books.google.co.uk/books?id=QDJNg5Nyef0C&pg=PR3#v=onepage&q&f=false] *Harry A Hoffner Jr and H Craig Melchert. A Grammar of the Hittite Language. [https://books.google.co.uk/books?id=Gq1QEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Edgar H Sturtevant and E Adelaide Hahn. A Comparative Grammar of the Hittite Language. 1951. [https://books.google.co.uk/books?id=5GRiAAAAMAAJ] *Jaan Puhvel. Hittite Etymological Dictionary. [https://books.google.co.uk/books?id=kghtOX_crPMC&pg=PP1#v=onepage&q&f=false] *Edgar H Sturtevant. A Hittite Glossary. 2nd Ed: 1936. Maltese *See [[w:mt:Bibljografija tal-lingwa Maltija]] Judaeo-Spanish (Ladino) *See [[w:lad:Vikipedya:Bibliografia del djudeo-espanyol]] Asian *Cliff Goddard. The Languages of East and Southeast Asia: An Introduction.2005. [https://books.google.co.uk/books?id=364UDAAAQBAJ&pg=PP1#v=onepage&q&f=false] South Asian *Kārumūri V Subbārāo. South Asian Languages: A Syntactic Typology. 2012. [https://books.google.co.uk/books?id=ZCfiGYvpLOQC&pg=PP1#v=onepage&q&f=false] *Veneeta Dayal and Anoop Mahajan. Clause Structure in South Asian Languages. 2004. [https://books.google.co.uk/books?id=puC-wWcl7tQC&pg=PP1#v=onepage&q&f=false] East Asian *Papers in East Asian Languages [https://books.google.co.uk/books?id=JIO5KcazJnYC] *Nam-kil Kim and Henry H Tiee. Studies in East Asian Linguistics. 1985. [https://books.google.co.uk/books?id=vxoaAQAAIAAJ] *Linguistic Interfaces in East-Asian Languages: A Festschrift in Honor of Yoshihisa Kitagawa. (Studies in East Asian Linguistics.) [https://books.google.co.uk/books?id=k8QYEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Benjamin A Elman (ed). Rethinking East Asian Languages, Vernaculars, and Literacies, 1000–1919. [https://books.google.co.uk/books?id=1Q6JBAAAQBAJ&pg=PR1#v=onepage&q&f=false] Chinese, Japanese and Korean *Reading in Asian Languages: Making Sense of Written Texts in Chinese, Japanese, and Korean. 2012. [https://books.google.co.uk/books?id=HZmpAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Insup Taylor and Martin M Taylor. Writing and Literacy in Chinese, Korean and Japanese. (Studies in Written Language and Literacy 3). 1995. [https://books.google.co.uk/books?id=WDw4gBaPjZgC&lpg=PP1&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=XW9IAAAAQBAJ&pg=PP1#v=onepage&q&f=false] Chinese and Japanese *Teruhiro Ishiguro and Kang Kwong Luke. Grammar in Cross-Linguistic Perspective: The Syntax, Semantics, and Pragmatics of Japanese and Chinese. (Linguistic insights, vol 57). Peter Lang. 2012. [https://books.google.co.uk/books?id=p1OqlAEACAAJ] Japan and Korea; Japanese and Korean *Nicolas Tranter (ed). The Languages of Japan and Korea. 2012. [https://books.google.co.uk/books?id=QB3DD8qSVnAC&pg=PP1#v=onepage&q&f=false] *Hajime Hoji (ed). Japanese/Korean Linguistics. 1990. vol 1. [https://books.google.co.uk/books?id=Suwm4WrQB6IC&pg=PP7#v=onepage&q&f=false] *Jieun Kiaer and Ben Cagan. Pragmatics in Korean and Japanese Translation. 2023. [https://books.google.co.uk/books?id=vnJ_EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *J Marshall Unger. The Role of Contact in the Origins of the Japanese and Korean Languages. University of Hawaii Press. 2009. [https://books.google.co.uk/books?id=sYULAQAAMAAJ] Japonic *Michinori Shimoji. An Introduction to the Japonic Languages: Grammatical Sketches of Japanese Dialects and Ryukyuan Languages. Brill. 2022. [https://books.google.co.uk/books?id=TO77EAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Yosuke Igarashi, Kenan Celik, Tatsuya Hirako and Hayato Aoi. Word-Prosodic Systems of Japonic Languages. Brill. 2026. [https://books.google.co.uk/books?id=B_3CEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Japanese and Ryukyuan *Moriyo Shimabukuro. The Accentual History of the Japanese and Ryukyuan Languages: A Reconstruction. 2007. [https://books.google.co.uk/books?id=n_V5DwAAQBAJ&pg=PR3#v=onepage&q&f=false] Japan *Masayoshi Shibatani. The Languages of Japan. CUP. 1990. [https://books.google.co.uk/books?id=sD-MFTUiPYgC&pg=PP1#v=onepage&q&f=false] *Handbook of Historical Japanese Linguistics [https://books.google.co.uk/books?id=xjz3EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Christopher Seeley. A History of Writing in Japan. 1991. [https://books.google.co.uk/books?id=KCZ2ya6cg88C&pg=PP1#v=onepage&q&f=false] Series *Handbooks of Japanese Language and Linguistics Ryukyuan *Handbook of the Ryukyuan Languages: History, Structure, and Use [https://books.google.co.uk/books?id=g_FeCAAAQBAJ&pg=PR3#v=onepage&q&f=false] Ainu *Handbook of the Ainu Language [https://books.google.co.uk/books?id=FAmKEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Katsunobu Izutsu. The Ainu Language: A Linguistic Introduction. Hokkaido University of Education. 2004. [https://books.google.co.uk/books?id=ty5kAAAAMAAJ] *Kirsten Refsing. The Ainu Language: The Morphology and Syntax of the Shizunai Dialect. 1986. [https://books.google.co.uk/books?id=LDJkAAAAMAAJ] *Batchelor. An Ainu-English-Japanese Dictionary. 1889: [https://books.google.co.uk/books?id=3gzhqi__TbEC&pg=PP7#v=onepage&q&f=false]. 2nd Ed: 1905: [https://archive.org/details/ainuenglishjapan00batcuoft/page/n4/mode/1up]. *Batchelor. A Grammar of the Ainu Language. 1903. [https://books.google.co.uk/books?id=G_xK9M0bOb8C] Japanese English *James Stanlaw. Japanese English: Language and Culture Contact. 2004. [https://books.google.co.uk/books?id=cdrHAQAAQBAJ&pg=PP1#v=onepage&q&f=false] ==Japanese== See [[Universal Bibliography/Languages/Japanese|Japanese]] ==Chinese== Chinese to English *英語大補帖~裝扮篇. Travelcom. (Taiwan). [https://books.google.co.uk/books?id=g7cWyH9k26EC&pg=PP1#v=onepage&q&f=false vol 9] [[Category:Languages]] ga8iqxbcwwbi4v9urh7zdt51wl0t902 2834956 2834955 2026-09-29T02:42:53Z James500 297601 /* */ Add 2834956 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of languages. World *Keith Brown and Sarah Ogilvie. Concise Encyclopedia of Languages of the World. Elsevier. 2009. [https://books.google.co.uk/books?id=F2SRqDzB50wC&pg=PP1#v=onepage&q&f=false] *Anatole V Lyovin, Brett Kessler and William R Leben. An Introduction to the Languages of the World. 2nd Ed: 2017: [https://books.google.co.uk/books?id=RQGTDQAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Asya Pereltsvaig. Languages of the World: An Introduction. 2012. [https://books.google.co.uk/books?id=8q06xer0vHkC&pg=PP1#v=onepage&q&f=false] *Merritt Ruhlen. A Guide to the World's Languages. Vol 1 (Classification). Stanford University Press. 1987. [https://books.google.co.uk/books?id=WAMbAAAAIAAJ] *Bernard Comrie. The World's Major Languages. 2nd Ed: 2009: [https://books.google.co.uk/books?id=9S0rDwAAQBAJ&pg=PP1#v=onepage&q&f=false]. *George L. Campbell and Gareth King. Compendium of the World's Languages. 3rd Ed: 2013: [https://books.google.co.uk/books?id=DWAqAAAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Martin D Joachim. Languages of the World: Cataloging Issues and Problems. 1993. [https://books.google.co.uk/books?id=6u18PtO0BoQC&pg=PP1#v=onepage&q&f=false] Origin *Roy Harris. Origin Of Language. 1996. [https://books.google.co.uk/books?id=386lU_0oUWoC&pg=PR3#v=onepage&q&f=false] *James R Hurford. Origins of Language: A Slim Guide. 2014. [https://books.google.co.uk/books?id=InTiAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Merritt Ruhlen. The Origin of Language: Tracing the Evolution of the Mother Tongue. 1994. [https://books.google.com/books?id=retrAAAAIAAJ] *Language Origin: A Multidisciplinary Approach. 1992. [https://books.google.co.uk/books?id=z_yPBAAAQBAJ&pg=PA1933#v=onepage&q&f=false] *Jürgen Trabant and Sean Ward (eds). New Essays on the Origin of Language. 2001. [https://books.google.co.uk/books?id=Pt501C6Zv94C&pg=PP1#v=onepage&q&f=false] *Claire Lefebvre, Bernard Comrie and Henri Cohen (eds). New Perspectives on the Origins of Language. Studies in Language Companion series, vol 144. [https://books.google.co.uk/books?id=S64bAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Denis Bouchard. The Nature and Origin of Language. 2013. [https://books.google.co.uk/books?id=4cRoAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Giorgio Fano. The Origins and Nature of Language. Indiana University Press. [https://books.google.com/books?id=fdlrAAAAIAAJ] *Jean Aitchison. The Seeds of Speech: Language Origin and Evolution. 1996. Canto Ed: 2000. [https://books.google.co.uk/books?id=68Y5gUavbzwC&pg=PP1#v=onepage&q&f=false] *Morris Swadesh. The Origin and Diversification of Language. 2006. 2017. [https://books.google.co.uk/books?id=klUPEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Prehistory *Rudolf Botha and Chris Knight (eds). The Prehistory of Language. 2009. [https://books.google.co.uk/books?id=36tLTfV_hLcC&pg=PP1#v=onepage&q&f=false] *G Révész. The Origins and Prehistory of Language. Longmans, Green and Co. [https://books.google.co.uk/books?id=GxRZAAAAMAAJ] History *Tore Janson. The History of Languages: An Introduction. 2012. [https://books.google.co.uk/books?id=pE2N7noPfEoC&pg=PP1#v=onepage&q&f=false] *Tore Janson. Speak: A Short History of Languages. 2002. [https://books.google.co.uk/books?id=mAgGOU2XmCAC&pg=PP1#v=onepage&q&f=false] *Nicholas Ostler. Empires of the Word: A Language History of the World. Preface dated 2004. [https://books.google.co.uk/books?id=Mz2kxr6v2X4C&pg=PP1#v=onepage&q&f=false] *Steven Roger Fischer. History of Language. 1999. [https://books.google.co.uk/books?id=5i1Ql7QQy0kC&pg=PP1#v=onepage&q&f=false] *A S Diamond. The History and Origin of Language. 1959: [https://books.google.co.uk/books?id=mjcGAQAAIAAJ]. Routledge Revivals. [https://books.google.co.uk/books?id=P5jiEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Henry Sweet. The History of Language. 1900. [https://books.google.co.uk/books?id=PC1GGpv7vlsC&pg=PR3#v=onepage&q&f=false] Social history *Peter Burke and Roy Porter (eds). The Social History of Language. 1987. [https://books.google.co.uk/books?id=oyRshxHVV5sC&pg=PP1#v=onepage&q&f=false] Story *Charles Barber. The Story of Language. Pan Books. 1964. [https://books.google.co.uk/books?id=gx0RAQAAIAAJ] *[[w:en:Mario Pei|Mario Pei]]. The Story of Language. 1949. Lippincott. Revised Ed: 1965. [https://books.google.co.uk/books?id=lqEviMzgv7wC]. Review: [https://books.google.co.uk/books?id=aaCvFv11ZJ4C 67] The Literary Guide 82 (May 1952) Classification *April McMahon and Robert McMahon. Language Classification by Numbers. 2005. [https://books.google.co.uk/books?id=CrEUDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *CF and FM Voegelin. Classification and Index of the World's Languages. (Foundations of Linguistics series). Elsevier. New York. 1977. ISBN 0444001557. [https://books.google.co.uk/books?id=2LAuAAAAYAAJ] Extinct *Johannes Friedrich. Extinct Languages. 1957. [https://books.google.co.uk/books?id=SzcDAAAAMAAJ] *K David Harrison. When Languages Die: The Extinction of the World's Languages and the Erosion of Human Knowledge. 2007. [https://books.google.co.uk/books?id=GTfRCwAAQBAJ&pg=PP1#v=onepage&q&f=false] Dead *Coulter H George. How Dead Languages Work. 2020. [https://books.google.co.uk/books?id=xEfWDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Indo-European languages *Mate Kapović (ed). The Indo-European Languages. 2nd Ed: 2017: [https://books.google.co.uk/books?id=8i0lDwAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Anna Giacalone Ramat and Paolo Ramat (eds). The Indo-European Languages. 1998. [https://books.google.co.uk/books?id=vwUMNCYbLL0C&pg=PP1#v=onepage&q&f=false] **La Lingue Indoeuropee. 1993. *Philip Baldi. An Introduction to the Indo-European Languages. 1983. [https://books.google.co.uk/books?id=lq-mkL23oh8C&pg=PP1#v=onepage&q&f=false] *W B Lockwood. A Panorama of Indo-European Languages. 1972. [https://books.google.co.uk/books?id=QTLMEQAAQBAJ&pg=PA1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=xJ0cAQAAIAAJ] Anatolian languages *Donald C Swanson. A Select Bibliography of the Anatolian Languages. 1948. [https://books.google.co.uk/books?id=OR3KP8kCjzUC] Reprinted from Bulletin of the New York Public Library, [https://books.google.co.uk/books?id=ktkaAAAAMAAJ vol 52], nos 5 and 6, May and June 1948, pp 3 to 26. Hittite *Theo van den Hout. The Elements of Hittite. 2011. [https://books.google.co.uk/books?id=QDJNg5Nyef0C&pg=PR3#v=onepage&q&f=false] *Harry A Hoffner Jr and H Craig Melchert. A Grammar of the Hittite Language. [https://books.google.co.uk/books?id=Gq1QEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Edgar H Sturtevant and E Adelaide Hahn. A Comparative Grammar of the Hittite Language. 1951. [https://books.google.co.uk/books?id=5GRiAAAAMAAJ] *Jaan Puhvel. Hittite Etymological Dictionary. [https://books.google.co.uk/books?id=kghtOX_crPMC&pg=PP1#v=onepage&q&f=false] *Edgar H Sturtevant. A Hittite Glossary. 2nd Ed: 1936. Maltese *See [[w:mt:Bibljografija tal-lingwa Maltija]] Judaeo-Spanish (Ladino) *See [[w:lad:Vikipedya:Bibliografia del djudeo-espanyol]] Asian *Cliff Goddard. The Languages of East and Southeast Asia: An Introduction.2005. [https://books.google.co.uk/books?id=364UDAAAQBAJ&pg=PP1#v=onepage&q&f=false] South Asian *Kārumūri V Subbārāo. South Asian Languages: A Syntactic Typology. 2012. [https://books.google.co.uk/books?id=ZCfiGYvpLOQC&pg=PP1#v=onepage&q&f=false] *Veneeta Dayal and Anoop Mahajan. Clause Structure in South Asian Languages. 2004. [https://books.google.co.uk/books?id=puC-wWcl7tQC&pg=PP1#v=onepage&q&f=false] East Asian *Papers in East Asian Languages [https://books.google.co.uk/books?id=JIO5KcazJnYC] *Nam-kil Kim and Henry H Tiee. Studies in East Asian Linguistics. 1985. [https://books.google.co.uk/books?id=vxoaAQAAIAAJ] *Linguistic Interfaces in East-Asian Languages: A Festschrift in Honor of Yoshihisa Kitagawa. (Studies in East Asian Linguistics.) [https://books.google.co.uk/books?id=k8QYEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Benjamin A Elman (ed). Rethinking East Asian Languages, Vernaculars, and Literacies, 1000–1919. [https://books.google.co.uk/books?id=1Q6JBAAAQBAJ&pg=PR1#v=onepage&q&f=false] Chinese, Japanese and Korean *Reading in Asian Languages: Making Sense of Written Texts in Chinese, Japanese, and Korean. 2012. [https://books.google.co.uk/books?id=HZmpAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Insup Taylor and Martin M Taylor. Writing and Literacy in Chinese, Korean and Japanese. (Studies in Written Language and Literacy 3). 1995. [https://books.google.co.uk/books?id=WDw4gBaPjZgC&lpg=PP1&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=XW9IAAAAQBAJ&pg=PP1#v=onepage&q&f=false] Chinese and Japanese *Teruhiro Ishiguro and Kang Kwong Luke. Grammar in Cross-Linguistic Perspective: The Syntax, Semantics, and Pragmatics of Japanese and Chinese. (Linguistic insights, vol 57). Peter Lang. 2012. [https://books.google.co.uk/books?id=p1OqlAEACAAJ] Japan and Korea; Japanese and Korean *Nicolas Tranter (ed). The Languages of Japan and Korea. 2012. [https://books.google.co.uk/books?id=QB3DD8qSVnAC&pg=PP1#v=onepage&q&f=false] *Hajime Hoji (ed). Japanese/Korean Linguistics. 1990. vol 1. [https://books.google.co.uk/books?id=Suwm4WrQB6IC&pg=PP7#v=onepage&q&f=false] *Jieun Kiaer and Ben Cagan. Pragmatics in Korean and Japanese Translation. 2023. [https://books.google.co.uk/books?id=vnJ_EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *J Marshall Unger. The Role of Contact in the Origins of the Japanese and Korean Languages. University of Hawaii Press. 2009. [https://books.google.co.uk/books?id=sYULAQAAMAAJ] Japonic *Michinori Shimoji. An Introduction to the Japonic Languages: Grammatical Sketches of Japanese Dialects and Ryukyuan Languages. Brill. 2022. [https://books.google.co.uk/books?id=TO77EAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Yosuke Igarashi, Kenan Celik, Tatsuya Hirako and Hayato Aoi. Word-Prosodic Systems of Japonic Languages. Brill. 2026. [https://books.google.co.uk/books?id=B_3CEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Japanese and Ryukyuan *Moriyo Shimabukuro. The Accentual History of the Japanese and Ryukyuan Languages: A Reconstruction. 2007. [https://books.google.co.uk/books?id=n_V5DwAAQBAJ&pg=PR3#v=onepage&q&f=false] Japan *Masayoshi Shibatani. The Languages of Japan. CUP. 1990. [https://books.google.co.uk/books?id=sD-MFTUiPYgC&pg=PP1#v=onepage&q&f=false] *Handbook of Historical Japanese Linguistics [https://books.google.co.uk/books?id=xjz3EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Christopher Seeley. A History of Writing in Japan. 1991. [https://books.google.co.uk/books?id=KCZ2ya6cg88C&pg=PP1#v=onepage&q&f=false] Series *Handbooks of Japanese Language and Linguistics Ryukyuan *Handbook of the Ryukyuan Languages: History, Structure, and Use [https://books.google.co.uk/books?id=g_FeCAAAQBAJ&pg=PR3#v=onepage&q&f=false] Ainu *Handbook of the Ainu Language [https://books.google.co.uk/books?id=FAmKEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Katsunobu Izutsu. The Ainu Language: A Linguistic Introduction. Hokkaido University of Education. 2004. [https://books.google.co.uk/books?id=ty5kAAAAMAAJ] *Kirsten Refsing. The Ainu Language: The Morphology and Syntax of the Shizunai Dialect. 1986. [https://books.google.co.uk/books?id=LDJkAAAAMAAJ] *Batchelor. An Ainu-English-Japanese Dictionary. 1889: [https://books.google.co.uk/books?id=3gzhqi__TbEC&pg=PP7#v=onepage&q&f=false]. 2nd Ed: 1905: [https://archive.org/details/ainuenglishjapan00batcuoft/page/n4/mode/1up]. *Batchelor. A Grammar of the Ainu Language. 1903. [https://books.google.co.uk/books?id=G_xK9M0bOb8C] Japanese English *James Stanlaw. Japanese English: Language and Culture Contact. 2004.[https://books.google.co.uk/books?id=cdrHAQAAQBAJ&pg=PP1#v=onepage&q&f=false] ==Japanese== See [[Universal Bibliography/Languages/Japanese|Japanese]] ==Chinese== Chinese to English *英語大補帖~裝扮篇. Travelcom. (Taiwan). [https://books.google.co.uk/books?id=g7cWyH9k26EC&pg=PP1#v=onepage&q&f=false vol 9] [[Category:Languages]] t76ywtua5e7eioc9gq6rng3ih5php24 0 330494 2834846 2834842 2026-09-28T12:16:30Z JSINST 3110286 2834846 wikitext text/x-wiki The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7). '''Background:''' Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref> and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs <ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials come from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref>[[File:Lifestyle Medicine Pillars.png|250px|right|<big> Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]] ===== <big>'''Prevalence studies''' </big> ===== [[/The Maritime Health database design // |1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 1.5 Excel recoding (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref> '''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations) <span lang="en" dir="ltr">'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</span> <big> 16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention Program</big> '''Organisation''' - [[/The John Snow Institute bylaws /]] - [[/|Manila conference Oct. 2026]] - General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain, Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== <references /> Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> t6rruio12kxnznm7ummhc0zmimutq0b 2834847 2834846 2026-09-28T12:22:17Z JSINST 3110286 2834847 wikitext text/x-wiki The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7). '''Background:''' Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref> and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs <ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials come from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref>[[File:Lifestyle Medicine Pillars.png|250px|right|<big> Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]] ===== <big>'''Prevalence studies''' </big> ===== [[/The Maritime Health database design // |1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 1.5 Excel recoding (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref> '''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations) <span lang="en" dir="ltr"><small>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</small></span> <small>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention Program</small> '''Organisation''' - [[/The John Snow Institute bylaws /]] - [[/|Manila conference Oct. 2026]] - General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain, Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== <references /> Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> du8jzrvri5ort2usl0w5449z1s0lbkr 2834848 2834847 2026-09-28T12:26:19Z JSINST 3110286 2834848 wikitext text/x-wiki The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7). '''Background:''' Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref> and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs <ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials come from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref>[[File:Lifestyle Medicine Pillars.png|250px|right|<big> Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]] ===== <big>'''Prevalence studies''' </big> ===== [[/The Maritime Health database design // |1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 1.5 Excel recoding (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref> '''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations) <span lang="en" dir="ltr"><small>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</small></span> <small>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref> Program</small> '''Organisation''' - [[/The John Snow Institute bylaws /]] - [[/|Manila conference Oct. 2026]] - General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain, Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== <references /> Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> a7j6rtgnlwcewy748u0zr5mcgm0dbna 2835024 2834848 2026-09-29T10:40:17Z JSINST 3110286 2835024 wikitext text/x-wiki The John Snow prediabetes Institute is an international research network focused on prediabetes remission, early risk identification, and metabolic health education. (https://w.wiki/Skm7). '''Background:''' Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight (General Research Draft). <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. The arteriosclerosis starts in the prediabetes stage. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention study''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref> and self-evaluation of diet and physical activity. Other intervention studies to be tested and developed include Digital lifestyle programs <ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials come from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref>[[File:Lifestyle Medicine Pillars.png|250px|right|<big> Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]] ===== <big>'''Prevalence studies''' </big> ===== [[/The Maritime Health database design // |1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 1.5 Excel recoding (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref> '''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations) <span lang="en" dir="ltr"><small>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</small></span> <small>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref> '''Organisation''' - [[/The John Snow Institute bylaws /]] - [[/|Manila conference Oct. 2026]] - General research pr[[Category:Prediabetes ]]otoco[[Category:Prediabetes ]]l draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Spain, Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== <references /> Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> g90q6vmf5cfs6qp4zho6jj6u5f1gt0n 0 331004 2835013 2834774 2026-09-29T09:30:47Z U3253363 3106362 Added a table to clearly communicate the UPPS-P model dimensions. 2835013 wikitext text/x-wiki {{title|Impulsivity versus sensation-seeking:<br>What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour?}} __TOC__ {{RoundBoxTop|theme=2}} [[File: Beer Cantus.jpg|right|thumb|250px|'''Figure 1.''' Social drinking can involve similar observable behaviours despite differences in the psychological processes underlying an individual's decision to participate.]] '''Imagine this...''' At a university party, two students are offered an unfamiliar alcoholic beverage. The first accepts immediately, without hesitation, considering the possible consequences. The second pauses but deliberately chooses to try it because the experience is new and exciting. From the outside, their behaviour looks almost identical: both accept the drink. Psychologically, however, the processes motivating their decisions may be quite different. {{RoundBoxBottom}} Impulsivity and sensation seeking are related but distinct psychological constructs that can help explain why people engage in seemingly similar behaviours for different reasons. Impulsivity is multidimensional and includes tendencies such as acting without sufficient forethought, whereas sensation seeking reflects the pursuit of novel, exciting, or stimulating experiences (Goh et al., 2020; Ravert & Donnellan, 2021). Consequently, risk-taking behaviour does not necessarily indicate that an individual is "impulsive". Understanding the motivation and psychological processes behind the behaviour can better explain why it occurs. Distinguishing impulsivity from sensation seeking is particularly important when considering risk-taking behaviour. Different dimensions of impulsivity and sensation seeking are associated with different patterns of health risk behaviour, suggesting that similar actions may arise through varied psychological pathways (Rogers et al., 2021). In the opening scenario, for instance, immediately accepting the drink without considering its consequences may reflect a lack of premeditation, whereas deliberately accepting it for novelty and excitement may more closely reflect sensation seeking. The behavioural outcome is similar, but the motivation is not. Alcohol use provides a useful real-world example of why this distinction matters. Research with university students suggests that sensation seeking and impulsivity can relate to alcohol involvement through different pathways, with sensation seeking more strongly associated with alcohol consumption and impulsivity more closely associated with alcohol related problems (Magid et al., 2007). Identifying these differences can therefore improve understanding not only of whether someone engages in risky behaviour but also of the psychological processes that may contribute to that behaviour and its consequences. This chapter examines the theoretical and empirical distinction between impulsivity and sensation seeking and evaluates how each may influence behaviour. By examining their conceptual overlap, underlying psychological processes, links to risk-taking, and associations with alcohol use, the chapter shows that understanding ''why'' a person behaves a certain way can be as important as observing ''what'' they do.{{RoundBoxTop|theme=3}} '''Focus questions''' * What is impulsivity and what are its key characteristics? * What is sensation seeking and how does it differ from impulsivity? * What psychological processes distinguish impulsivity from sensation seeking? * How do impulsivity and sensation seeking differentially influence risk-taking behaviour? * How can these differences help explain alcohol use and related consequences? {{RoundBoxBottom}} == Understanding impulsivity == Impulsivity is commonly understood as acting without thinking, but psychological research suggests that it is not a single characteristic. Behaviours described as impulsive can result from varied psychological processes, such as acting impetuously during intense emotions, failing to consider future consequences, or struggling to persist with difficult tasks. Research supports this multidimensional understanding, finding that measures labelled as impulsivity assess several related but distinguishable characteristics rather than one unified trait (Berg et al., 2015; Sharma et al., 2014). Thus, understanding why an individual acts impulsively may provide more information than simply labelling the behaviour as impulsive. === The UPPS-P model === One influential framework for understanding these differences is the UPPS-P model of impulsive personality. The original UPPS model proposed four dimensions: negative urgency, or acting rashly during negative emotion; lack of premeditation, or acting without adequately considering consequences; lack of perseverance, involving difficulty remaining focused on demanding or boring tasks; and sensation seeking, involving a preference for exciting and novel experiences (see Table 1; Whiteside & Lynam, 2001). The framework was later expanded to distinguish positive urgency, referring to hasty actions during intense positive emotion, from negative urgency (Cyders & Smith, 2008). Consequently, two apparently impulsive decisions may originate from quite different combinations of emotion, planning, persistence and reward seeking. '''Table 1''' ''UPPS Dimensions'' {| class="wikitable" |+ !Model name acronym !Dimension !Explanation |- |U |Negative urgency |Acting rashly during negative emotion |- |P |Lack of premeditation |Acting without adequately considering consequences |- |P |Lack of perseverance |Difficulty remaining focused on demanding or boring tasks |- |S |Sensation seeking |A preference for exciting and novel experiences |- | - P |Positive urgency |Hasty actions during intense positive emotion |} Experts agree that these components should be measured individually. Network analysis of the UPPS-P found meaningful relationships among its components and indicated that the dimensions are not interchangeable (Goh et al., 2020). Other research has continued to examine and refine multidimensional measures of impulsivity, demonstrating both the usefulness and continuing complexity of defining the construct (Samiefard et al., 2023; Türkmen et al., 2023). This is an important limitation of the broad term impulsivity: although multidimensional models provide greater precision, disagreement remains about how impulsivity should be organised and measured. === Emotion and urgency === [[File:Positive & Negative Urgency.png|right|thumb|400px|'''Figure 2'''. Positive and negative urgency describe tendencies towards rash action during intense positive and negative emotional states. Although the emotional states differ, both illustrate how intense emotion may contribute to impulsive action (Cyders & Smith, 2008).]] The distinction between positive and negative urgency further demonstrates why motivation matters. Urgency theory proposes that intense emotional states can increase the likelihood of reckless action, with positive urgency occurring during strongly positive emotion and negative urgency during strongly negative emotion (Cyders & Smith, 2008). This differs from lack of premeditation because a person may ordinarily consider consequences effectively but behave differently when experiencing intense emotion. The opening party scenario illustrates this distinction. Immediately accepting the unfamiliar drink without adequately considering its consequences may indicate lack of premeditation. If the decision instead occurred during intense excitement at the party, positive urgency might provide another explanation. The observable action alone, therefore, cannot identify which dimension was responsible. <quiz display=simple> {According to the UPPS-P model, impulsivity is best understood as a single characteristic that affects behaviour in the same way across different situations.} - True + False {Positive urgency refers to rash action during intense positive emotion, whereas negative urgency refers to rash action during intense negative emotion.} + True - False </quiz> == Understanding sensation seeking == Sensation seeking is strongly correlated with impulsivity but emphasises the motivation to experience novelty, intensity, and stimulation. The construct has traditionally been understood as an individual difference in preference for varied, novel, complex, and intense experiences, sometimes accompanied by a willingness to accept risk to obtain them (Roberti, 2004; Zuckerman & Kuhlman, 2000). This ultimately indicates that risk itself does not always have to be the person's goal. Instead, people may accept risk because it accompanies an experience they find sufficiently interesting or rewarding. === A motivational perspective === This distinguishes it from other dimensions of impulsivity. Someone displaying a lack of premeditation may act before adequately considering the consequences, whereas a sensation seeker may recognise the consequences and deliberately decide that the potential stimulation is worthwhile. In the opening scenario, the second student hesitates before accepting the unfamiliar drink but chooses it because the experience is new and exciting. The decision may therefore involve forethought while still carrying risk. Sensation seeking should consequently not be treated as inherently dysfunctional. Ravert and Donnellan (2021) found different associations between impulsivity, sensation seeking, and psychological well-being, supporting the idea that the constructs should not simply be combined into a general tendency toward problematic behaviour. Sensation seeking may encourage exploration and engagement as well as potentially hazardous behaviour, meaning that its consequences depend partly on how and where the need for stimulation is expressed. === A theoretical complication === A key theoretical complication is that sensation seeking is treated as both a distinct construct and a dimension within the UPPS-P model of impulsivity (Whiteside & Lynam, 2001). Including sensation seeking within the UPPS-P model does not mean it is the same as other forms of impulsivity; rather, it is considered a distinct pathway that may contribute to impulsive behaviour. In contrast, it shows that the boundaries around the broader concept of impulsivity remain debated and depend partly on how researchers define and measure it. If all sensation seeking were simply impulsivity, there would be little reason to expect the two to show different relationships with behaviour. Evidence that they predict different outcomes can therefore help determine whether distinguishing them is psychologically meaningful. <quiz display=simple> {Which statement best distinguishes sensation seeking from lack of premeditation?} - Sensation seeking always involves acting without considering the consequences. + Sensation seeking may involve recognising potential consequences but deciding that the novelty or stimulation is worthwhile. - Lack of premeditation involves deliberately pursuing novel and exciting experiences. - Sensation seeking and lack of premeditation describe the same psychological process. {Why is sensation seeking theoretically complicated within models of impulsivity?} - Sensation seeking has no relationship with impulsive behaviour. - Sensation seeking has been removed from multidimensional models of impulsivity. + Sensation seeking can be treated as a distinct construct while also being included as a dimension within the UPPS-P model. - Sensation seeking only occurs when an individual does not recognise the risks of their behaviour. </quiz> == Distinguishing impulsivity from sensation seeking == [[File:Sensation seeking Vs Impulsivity.png|right|thumb|387px|'''Figure 3'''. Comparison of the UPPS-P dimensions of impulsive behaviour and Zuckerman’s dimensions of sensation seeking, highlighting areas of conceptual overlap and distinction (Goh et al., 2020; Magid et al., 2007; Samiefard et al., 2023). (image created independently using Canva software).]] Impulsivity and sensation seeking overlap because both can contribute to behaviours involving uncertainty or risk. The key distinction concerns the process motivating the action. Dimensions such as urgency and lack of premeditation emphasise rash action or poor regulation, whereas sensation seeking emphasises an approach to novelty, excitement, and stimulation (Cyders & Smith, 2008; Zuckerman & Kuhlman, 2000). Similar behaviour can therefore emerge through different psychological pathways. Research comparing these constructs supports this distinction. Magid et al. (2007) found that sensation seeking and impulsivity showed different relationships with alcohol involvement, suggesting that combining them could overlook meaningful differences. Similarly, a meta-analysis found that sensation seeking and impulsivity were both associated with greater risk-taking, but sensation seeking showed a somewhat stronger relationship with risky behaviour than impulsivity (Lauriola et al., 2014). These findings support distinguishing the constructs while also showing that neither fully explains risky behaviour. '''Table 1''' '''Table – Comparing impulsivity and sensation''' '''seeking''' The comparison table summarises the major conceptual differences and similarities between impulsivity and sensation seeking, including their defining characteristics, underlying processes, and relationships with risk taking behaviour. {| class="wikitable" style="margin: auto; |- ! Feature !! Impulsivity !! Sensation seeking |- | '''Core characteristic''' || Tendency toward rash or insufficiently considered action across multiple dimensions || Tendency to seek novel, varied, exciting or stimulating experiences |- | '''Key Process''' || Can involve reduced forethought, urgency or behavioural regulation || Motivation toward novelty, excitement and stimulation |- |'''Risk Taking''' |Risk may result from insufficient consideration of consequences or other impulsive tendencies |Risk may be accepted in pursuit of rewarding or stimulating experiences |- |'''UPPS-P''' |Multidimensional model including positive urgency, negative urgency, lack of premeditation, lack of perseverance and sensation seeking |Sensation seeking is one distinct dimension within the UPPS-P model |- |'''Example''' |Immediately accepting an unfamiliar drink without considering consequences |Deliberately trying the drink because the experience is novel and exciting |- |'''Behavioural Outcome''' |Can contribute to health risk, gambling and substance related behaviours |Can contribute to risk taking, alcohol use and novelty oriented behaviour |} Whilst impulsivity and sensation seeking can contribute to similar observable behaviours, their underlying characteristics and psychological processes differ (see Table 1; Goh et al., 2020; Magid et al., 2007). === Development and the dual systems model === The dual systems model offers another theoretical perspective on this distinction. The model proposes that heightened reward sensitivity develops differently from capacities associated with cognitive control, potentially contributing to increased risk-taking during adolescence (Steinberg, 2010). From this perspective, attraction towards rewarding or stimulating experiences and the ability to regulate behaviour are related but separable processes. Conversely, the model has also been critically reconsidered. Shulman et al. (2016) concluded that evidence broadly supports different developmental trajectories for reward-related and cognitive control systems, while also highlighting complexities that a simple imbalance between the two systems cannot capture. Risk-taking varies substantially across people and situations, so developmental models should not be interpreted as suggesting that risky behaviour is inevitable or caused by one neural mechanism. The theory helps explain why reward seeking and behavioural control can be distinguished, but it should complement rather than replace personality and contextual explanations. == Impulsivity, sensation seeking, and risk taking behaviour == [[File:Boxing in Uruguay - Palacio Peñarol.jpg|right|thumb|350px|'''Figure 4'''. Activities involving risk and stimulation, such as boxing, illustrate that engagement in risky behaviour does not necessarily indicate impulsivity. The psychological motivation underlying the behaviour may be important for distinguishing impulsivity from sensation seeking.]] The theoretical distinction is particularly useful when considering risk-taking behaviour. If impulsivity and sensation seeking represented the same underlying characteristic, they would be expected to show broadly equivalent relationships with risky outcomes. Instead, evidence indicates that different dimensions can be associated with different behaviours and consequences. Research examining risk-taking has found that both sensation seeking and impulsivity are associated with a greater tendency to engage in risky behaviour, although these relationships are relatively modest (Lauriola et al., 2014). This pattern extends across behaviour types, with sensation seeking and impulsivity-related traits associated with health risk behaviours (Rogers et al., 2021), adolescent risk-taking (Siraj et al., 2021), and speeding behaviour (Sârbescu & Rusu, 2021). More recently, Grubbs et al. (2024) demonstrated that the specific psychological trait involved may also relate to different aspects of risky behaviour. Sensation seeking was associated with whether individuals had engaged in sports wagering, while urgency-related dimensions were associated with different patterns of wagering involvement and gambling-related problems. These findings support the theoretical distinction but require cautious interpretation. Personality traits generally explain only part of the variation in complex behaviours, and correlations cannot establish that a trait directly causes an individual to take a particular risk. Social influences, opportunity, emotional state, perceived rewards and the specific type of risk may interact with personality characteristics. Therefore, describing someone simply as a "risk taker" may obscure both the psychological pathway and the context contributing to their behaviour. == Alcohol use and related consequences == Alcohol use provides an especially useful real-world example because drinking and experiencing alcohol related problems are not the same outcome. Sensation seeking may motivate approach towards stimulating social or drinking experiences, whereas other impulsivity dimensions may be more relevant to rash decisions and harmful consequences. This distinction directly reflects the opening scenario: both students accept the same drink; however, their reasons for doing so differ. Magid et al. (2007) found that sensation seeking and impulsivity related differently to alcohol use and alcohol related problems among university students. Broader research evidence supports this outcome-specific approach; for instance, across 96 studies, Coskunpinar et al. (2013) found that different UPPS dimensions showed different relationships with alcohol outcomes, with urgency dimensions particularly relevant to alcohol related problems. Evidence from individuals entering residential alcohol detoxification also demonstrates clinically relevant relationships between UPPS-P dimensions and substance-related characteristics (Kempeneers et al., 2023). These findings illustrate why psychological explanations should expand beyond asking whether someone is "impulsive". Identifying whether behaviour is motivated primarily by stimulation, emotional urgency, insufficient forethought, or another process may provide a more precise understanding of the behaviour. At the same time, much of this evidence is correlational. It relies on self-report measures, so personality should not be interpreted as determining whether an individual will drink or experience alcohol related harm. === From explanation to application === Understanding different pathways may also have practical implications. Individuals motivated primarily by sensation seeking may benefit from finding safer ways to obtain novelty and stimulation. In contrast, behaviour associated with urgency may require greater attention to managing intense emotional states before acting. Lack of premeditation is particularly relevant; strategies that build in time to consider consequences may be more appropriate. These possibilities illustrate why distinguishing psychological processes can improve everyday behaviour rather than merely attaching personality labels to people. However, individualised strategies should not be inferred solely from personality scores, and the available evidence does not demonstrate that matching interventions to UPPS-P dimensions will necessarily produce better outcomes. Instead, the distinction provides a framework for understanding why the same behaviour may require different explanations and potentially different approaches to change. '''Summary Quiz: Impulsivity or sensation seeking?'''<quiz display="simple"> {A student is offered an unfamiliar alcoholic drink at a party and immediately accepts without considering the possible consequences. Which characteristic best explains this behaviour? |type="()"} + Lack of premeditation - Thrill and adventure seeking - Experience seeking - Boredom susceptibility {A student deliberately chooses to try an unfamiliar activity because they enjoy novelty, excitement, and stimulation. Which construct best explains this motivation? |type="()"} - Lack of perseverance + Sensation seeking - Negative urgency - Lack of premeditation {Which statement best describes the distinction between impulsivity and sensation seeking? |type="()"} - Impulsivity and sensation seeking are interchangeable terms for the same psychological construct. - Sensation seeking always involves acting without considering consequences. + Similar observable behaviours can arise from different underlying motivations and psychological processes. - Only impulsivity is associated with risk taking behaviour. </quiz> == Conclusion == Impulsivity and sensation seeking are related psychological constructs; however, they should not automatically be treated as interchangeable. Multidimensional models such as the UPPS-P show that behaviour commonly labelled "impulsive" can arise through different pathways involving emotion, forethought, persistence, and attraction to stimulation. Sensation seeking is particularly important because it overlaps conceptually with impulsivity while emphasising the motivation to pursue novel and exciting experiences. Research supports the practical value of distinguishing these processes. Sensation seeking and different dimensions of impulsivity show different patterns of association with risk-taking, alcohol involvement and other potentially harmful behaviours. However, these relationships are generally probabilistic rather than deterministic. Consequently, observable behaviour alone cannot reveal why an individual acted as they did. The opening scenario illustrates the practical significance of distinguishing between these psychological processes. Accepting an unfamiliar drink may result from acting without considering consequences, seeking an exciting new experience, responding rashly to intense emotion, or a combination of these processes. Understanding why behaviour occurs, rather than simply labelling what occurred, can provide a more useful foundation for understanding and potentially improving everyday decision-making. == See also == * [[Motivation and emotion/Book/2011/Sensation seeking|Sensation seeking]] (Book chapter, 2011) * [[w:Sensation Seeking Scale|Sensation Seeking Scale]] (Wikipedia) == References == {{Hanging indent|1= Berg, J. M., Latzman, R. D., Bliwise, N. G., & Lilienfeld, S. O. (2015). Parsing the heterogeneity of impulsivity: A meta-analytic review of the behavioral implications of the UPPS for psychopathology. ''Psychological Assessment, 27''(4), 1129–1146. https://doi.org/10.1037/pas0000111 Coskunpinar, A., Dir, A. L., & Cyders, M. A. (2013). Multidimensionality in impulsivity and alcohol use: A meta-analysis using the UPPS model of impulsivity. ''Alcoholism: Clinical and Experimental Research, 37''(9), 1441–1450. https://doi.org/10.1111/acer.12131 Cyders, M. A., & Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin, 134''(6), 807–828. https://doi.org/10.1037/a0013341 Goh, P. K., Lee, C. A., Martel, M. M., Fillmore, M. T., Derefinko, K. J., & Lynam, D. R. (2020). Conceptualizing the UPPS-P model of impulsive personality through network analysis: Key dimensions and general robustness across young adulthood. ''Journal of Personality, 88''(6), 1302–1314. https://doi.org/10.1111/jopy.12572 Grubbs, J. B., Connolly, A. J., Floyd, C. G., Kim, H. S., & Kraus, S. W. (2024). Impulsivity and diverse forms of sports wagering in the U.S.: An examination of the UPPS-P model. ''Addictive Behaviors, 156'', 108058. https://doi.org/10.1016/j.addbeh.2024.108058 Kempeneers, P., Mreyen, K., Pallincourt, R., Remacle, F., Wildemeersch, G., & Simon, J. (2023). Validation of the UPPS-P Impulsive Behavior Scale and clinical correlates of its scores in French-speaking patients starting a residential detoxification program. ''Indian Journal of Psychological Medicine, 45''(5), 503–510. https://doi.org/10.1177/02537176231157411 Lauriola, M., Panno, A., Levin, I. P., & Lejuez, C. W. (2014). Individual differences in risky decision making: A meta-analysis of sensation seeking and impulsivity with the Balloon Analogue Risk Task. ''Journal of Behavioral Decision Making, 27''(1), 20–36. https://doi.org/10.1002/bdm.1784 Magid, V., MacLean, M. G., & Colder, C. R. (2007). Differentiating between sensation seeking and impulsivity through their mediated relations with alcohol use and problems. ''Addictive Behaviors, 32''(10), 2046–2061. https://doi.org/10.1016/j.addbeh.2007.01.015 Ravert, R. D., & Donnellan, M. B. (2021). Impulsivity and sensation seeking: Differing associations with psychological well-being. ''Applied Research in Quality of Life, 16''(4), 1503–1515. https://doi.org/10.1007/s11482-020-09829-y Roberti, J. W. (2004). A review of behavioral and biological correlates of sensation seeking. ''Journal of Research in Personality, 38''(3), 256–279. https://doi.org/10.1016/S0092-6566(03)00067-9 Rogers, M. M., Kelley, K., & McKinney, C. (2021). Trait impulsivity and health risk behaviors: A latent profile analysis. ''Personality and Individual Differences, 171'', 110511. https://doi.org/10.1016/j.paid.2020.110511 Samiefard, M., Salehi Fadardi, J., Kareshki, H., & Stacy, A. W. (2023). Validity and reliability of a revised S-UPPS-P Impulsive Behavior Scale: The interplay between impulsivity and working memory. ''Journal of Personality Assessment, 105''(2), 174–186. https://doi.org/10.1080/00223891.2022.2081922 Sârbescu, P., & Rusu, A. (2021). Personality predictors of speeding: Anger-aggression and impulsive-sensation seeking. A systematic review and meta-analysis. ''Journal of Safety Research, 77'', 86–98. https://doi.org/10.1016/j.jsr.2021.02.004 Sharma, L., Markon, K. E., & Clark, L. A. (2014). Toward a theory of distinct types of “impulsive” behaviors: A meta-analysis of self-report and behavioral measures. ''Psychological Bulletin, 140''(2), 374–408. https://doi.org/10.1037/a0034418 Shulman, E. P., Smith, A. R., Silva, K., Icenogle, G., Duell, N., Chein, J., & Steinberg, L. (2016). The dual systems model: Review, reappraisal, and reaffirmation. ''Developmental Cognitive Neuroscience, 17'', 103–117. https://doi.org/10.1016/j.dcn.2015.12.010 Siraj, R., Najam, B., & Ghazal, S. (2021). Sensation seeking, peer influence, and risk-taking behavior in adolescents. ''Education Research International, 2021'', 8403024. https://doi.org/10.1155/2021/8403024 Steinberg, L. (2010). A dual systems model of adolescent risk-taking. ''Developmental Psychobiology, 52''(3), 216–224. https://doi.org/10.1002/dev.20445 Türkmen, İ., Rodoplu, N., Üner, B. S., Esmer, Ş. C., Altan-Atalay, A., & Ece, B. (2023). When the UPPS-P model of impulsivity meets a revised approach: The development and validation of the TRUE Multidimensional Impulsivity Scale. ''Journal of Personality Assessment, 105''(3), 355–370. https://doi.org/10.1080/00223891.2022.2093730 Whiteside, S. P., & Lynam, D. R. (2001). The five factor model and impulsivity: Using a structural model of personality to understand impulsivity. ''Personality and Individual Differences, 30''(4), 669–689. https://doi.org/10.1016/S0191-8869(00)00064-7 Zuckerman, M., & Kuhlman, D. M. (2000). Personality and risk-taking: Common biosocial factors. ''Journal of Personality, 68''(6), 999–1029. https://doi.org/10.1111/1467-6494.00124 }} ==External links== * [https://dictionary.apa.org/sensation-seeking-scale APA Dictionary of Psychology – Sensation-Seeking Scale] * [https://onlinelibrary.wiley.com/doi/full/10.1002/9781405186407.wbiecs029 Sensation Seeking – Zuckerman] {{ic|Move academic sources to citations/references}} * [https://www.impulsivity.org/measurement/upps_p/ UPPS-P Impulsive Behavior Scale] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Impulsivity]] [[Category:Motivation and emotion/Book/Sensation seeking]] 3b29sf7e1oky7byo7dz1wbondqxczf7 828 331015 2834930 2834744 2026-09-28T23:40:50Z Helpme2222 3106525 2834930 Scribunto text/plain -- My function for teaching grammar local p = {}; local frame = mw.getCurrentFrame() local grammatical_moods = { "Jelen ido", "Mult ido", "Felszolito mod", "Felteteles mod" } local balls = { grammatical_moods } local toggleButton = mw.html.create( 'span' ) :css({ float = "right", ["margin-right"] = "0", ["font-weight"] = "normal" }) :addClass("mw-customtoggle-Infobox") :node("[Click to toggle]") function p.subheading( discrim, b_content, content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node((discrim) and b_content or content) :node((discrim) and toggleButton) ) end function p.s_subheading( s_content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible mw-collapsed" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node(s_content) :node(toggleButton) ) end function p.row( discrim, name, data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(name) ) :node( mw.html.create( 'td' ) :node(data) ) end function p.s_row( s_name, s_data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass("mw-collapsible mw-collapsed") :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(s_name) ) :node( mw.html.create( 'td' ) :node(s_data) ) end function p.var(symbol) return frame:expandTemplate{ title = "Template:Var", args = {symbol} } end function p.iseq() return table.concat({ frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/'") and " is equal to User:Helpme2222/Sandbox/'" or " is not equal to User:Helpme2222/Sandbox/'"), frame.args[2] .. ( (frame.args[2] == "User:Helpme2222/Sandbox/&#039;") and " is equal to User:Helpme2222/Sandbox/&#039;" or " is not equal to User:Helpme2222/Sandbox/&#039;"), frame.args[3] .. ( (frame.args[3] == "Helpme2222/Sandbox/'") and " is equal to Helpme2222/Sandbox/'" or " is not equal to Helpme2222/Sandbox/'"), frame.args[4] .. ( (frame.args[4] == "Helpme2222/Sandbox/&#039;") and " is equal to Helpme2222/Sandbox/&#039;" or " is not equal to Helpme2222/Sandbox/&#039;"), }) end function p.infoboxWBB() -- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html --[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node Thankfully, these operations interpret passed nils as no-ops (I'm technically using false though. Somehow that works anyways?)]] local frame = mw.getCurrentFrame() local ballname = frame.args["Title"] or tostring(mw.title.getCurrentTitle()) local Entities = p -- require("Module:Entities") local featureCount = tonumber(frame.args["Feature count"]) or 1 local s_featureCount = tonumber(frame.args["SUPER Feature count"]) or featureCount local matchCount = { total = mw.site.stats.pagesInCategory( ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local matchWinCount = { standard = mw.site.stats.pagesInCategory( ballname.."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local s_matchCount = { total = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local s_matchWinCount = { standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local isSuper = frame.args["SUPER Feature count"] or frame.args["SUPER Type"] or frame.args["SUPER Debuted"] or (s_matchCount.total > 0) or nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra "or nil" had to be added since this is used in :addClass calls. -- TITLE local infobox = mw.html.create( 'table' ) :addClass("wikitable") :css({ float = "right"; ["margin-left"] = "10px" }) :node( mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass((isSuper) and "mw-collapsible" or nil) :node("<big>" .. ballname .. "</big>") ) ) :node((isSuper) and mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass("mw-collapsible mw-collapsed") :node("<big>" .. (frame.args["SUPER Title"] or ("Super " .. ballname)) .. "</big>") ) ) -- SECTION: Identification -- Section Heading :node( p.subheading( (frame.args["SUPER Type"]), "Base Identification", "Basic Info" ) ) :node( (frame.args["SUPER Type"]) and p.s_subheading( "Super Identification" ) ) -- Type :node( p.row( (frame.args["SUPER Type"]), "Type", frame.args["Type"] ) ) :node( (frame.args["SUPER Type"]) and p.s_row( "Type", frame.args["SUPER Type"] ) ) -- Color local color = frame.args["Color"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "color" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Color") ) :node( mw.html.create( 'td' ) :node( (color ~= "") and frame:expandTemplate{ title = "Color", args = {color} } or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add color...</i>]]" ) ) ) -- Emoji local emoji = frame.args["Emoji"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "emoji" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Emoji") ) :node( mw.html.create( 'td' ) :node( (emoji ~= "") and ( emoji .. " <code>" .. p.emojiCodepointInternal(emoji) .. "</code>") or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add emoji...</i>]]" ) ) ) -- SECTION: Scaling Characteristics --[[ Some data will be collected first. ]] -- (Base) Behavior local features = {} local starts = {} local scalings = {} if (featureCount > 1) then for i = 1, featureCount do table.insert(features, { frame.args["Feature "..i], " (" .. p.var(frame.args["Var "..i]) .. ")" }) table.insert(starts, { p.var( frame.args["Var0 "..i] or (frame.args["Var "..i].."₀") ) .. ": ", frame.args["Starting value "..i] }) table.insert(scalings, { p.var( "Δ"..(frame.args["Var "..i]) ) .. ": ", frame.args["Scaling "..i] }) end else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1 features[1] = {frame.args["Feature"],""} starts[1] = {"",frame.args["Starting value"]} scalings[1] = {"",frame.args["Scaling"]} end -- Super Behavior local s_features local s_starts local s_scalings -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default. local has_s_features; local has_s_starts; local has_s_scalings; if (frame.args["SUPER Feature count"]) then if s_featureCount > 1 then -- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data s_features = mw.clone(features) s_starts = mw.clone(starts) s_scalings = mw.clone(scalings) -- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors. if s_featureCount > featureCount then for i = featureCount + 1, s_featureCount do table.insert( s_features, {"",""} ) table.insert( s_starts, {"",""} ) table.insert( s_scalings, {"",""} ) end elseif s_featureCount < featureCount then for i = s_featureCount + 1, featureCount do s_features[i] = nil s_starts[i] = nil s_scalings[i] = nil end end -- Parse super data, set the has_s_... flags if triggered for i = 1,s_featureCount do if frame.args["SUPER Feature "..i] then s_features[i][1] = frame.args["SUPER Feature "..i] if (not has_s_features) then has_s_features = true end end if frame.args["SUPER Starting value "..i] then s_starts[i][2] = frame.args["SUPER Starting value "..i] if (not has_s_starts) then has_s_starts = true end end if frame.args["SUPER Scaling "..i] then s_scalings[i][2] = frame.args["SUPER Scaling "..i] if (not has_s_scalings) then has_s_scalings = true end end if frame.args["SUPER Var "..i] then s_features[i][2] = " (" .. p.var(frame.args["SUPER Var "..i]) .. ")" s_starts[i][1] = p.var( frame.args["SUPER Var0 "..i] or (frame.args["SUPER Var "..i].."₀") ) .. ": " s_scalings[i][1] = p.var( '<span style="font-style:normal">Δ</span>'..(frame.args["SUPER Var "..i]) ) .. ": " if (not has_s_features) then has_s_features = true end if (not has_s_starts) then has_s_starts = true end if (not has_s_scalings) then has_s_scalings = true end end s_features[i] = table.concat(s_features[i]) s_starts[i] = table.concat(s_starts[i]) s_scalings[i] = table.concat(s_scalings[i]) end s_features = table.concat(s_features,"<br>") s_starts = table.concat(s_starts,"<br>") s_scalings = table.concat(s_scalings,"<br>") else -- # of super features = 1? just write directly to the variable if frame.args["SUPER Var"] or frame.args["SUPER Feature"] then has_s_features = true s_features = frame.args["SUPER Feature"] end if frame.args["SUPER Var0"] or frame.args["SUPER Var"] or frame.args["SUPER Starting value"] then has_s_starts = true s_starts = frame.args["SUPER Starting value"] end if frame.args["SUPER Scaling"] then has_s_scalings = true s_scalings = frame.args["SUPER Scaling"] end end end -- finalize base data as strings now that work is otherwise complete if (featureCount > 1) then for i = 1, featureCount do features[i] = table.concat(features[i]) starts[i] = table.concat(starts[i]) scalings[i] = table.concat(scalings[i]) end features = table.concat(features,"<br>") starts = table.concat(starts,"<br>") scalings = table.concat(scalings,"<br>") else features = features[1][1] starts = starts[1][2] scalings = scalings[1][2] end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (frame.args["SUPER Feature count"]), "Base Scaling Characteristics", "Scaling Characteristics" ) ) :node( (frame.args["SUPER Feature count"]) and p.s_subheading( "Super Scaling Characteristics" ) ) -- Features :node( p.row( (has_s_features), (featureCount > 1) and "Features" or ( (frame.args["Var"]) and ("Feature (" .. p.var(frame.args["Var"]) .. ")") or "Feature" ), features ) ) :node( (has_s_features) and p.s_row( (s_featureCount > 1) and "Features" or ( (frame.args["SUPER Var"]) and ("Feature (" .. p.var(frame.args["SUPER Var"]) .. ")") or "Feature" ), s_features ) ) -- Starting values :node( p.row( (has_s_starts), (featureCount > 1) and "Starting values" or ( (frame.args["Var"]) and ("Starting value (" .. ( p.var(frame.args["Var0"] or (frame.args["Var"].."₀") ) ) .. ")") or "Starting value" ), starts ) ) :node( (has_s_starts) and p.s_row( (s_featureCount > 1) and "Starting values" or ( (frame.args["SUPER Var"]) and ("Starting value (" .. ( p.var(frame.args["SUPER Var0"] or (frame.args["SUPER Var"].."₀") ) ) .. ")") or "Starting value" ), s_starts ) ) -- Scaling :node( p.row( (has_s_scalings), "Scaling", scalings) ) :node( (has_s_scalings) and p.s_row("Scaling", s_scalings) ) -- SECTION: Career --[[ Some data will be collected first. ]] -- (Base) Games played local gamesPlayedText = {} if matchCount.total > 0 then table.insert(gamesPlayedText, matchCount.total .. " (total)") if matchCount.standard > 0 then table.insert(gamesPlayedText, matchCount.standard .. " (standard)") end end -- (Base) Winrate local winrateText = {} if matchCount.total > 0 then if matchCount.standard > 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. "% (standard)" ) end end -- Super Games played local s_gamesPlayedText = {} if s_matchCount.total > 0 then table.insert(s_gamesPlayedText, s_matchCount.total .. " (total)") if s_matchCount.standard > 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. " (standard)") end end -- Super Winrate local s_winrateText = {} if s_matchCount.total > 0 then if s_matchCount.standard > 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. "% (standard)" ) end end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (isSuper), "Base Career", "Career ") ) :node( (isSuper) and p.s_subheading( "Super Career") ) -- Debuted :node( p.row( (isSuper), "Debut", frame.args["Debut"] ) ) :node( (isSuper) and p.s_row("Debut", frame.args["SUPER Debut"]) ) -- Games played :node( p.row( (isSuper), "Games played", (#gamesPlayedText > 0) and table.concat(gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Games played", (#s_gamesPlayedText > 0) and table.concat(s_gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) -- Winrate :node( p.row( (isSuper), "Winrate", (#winrateText > 0) and table.concat(winrateText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Winrate", (#s_winrateText > 0) and table.concat(s_winrateText, "<br>") or "<i>Not enough data...</i>" ) ) return tostring(infobox) end function p.referenceTest() local element = mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible" ) :css( "text-align","left" ) local bigElement = mw.html.create( "div" ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface") ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface2") ) return tostring(element) .. "\n" .. tostring(bigElement) .. "\n" end function p.roundedPct(num, den) return (num/den % 0.01 < 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100) end function p.title() --[[local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA" } } local cases2 = "ba" local cases3 = "ba" -- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = "true" else cases2 = "false" end -- if "ANCOVA" == frame:getParent():getTitle() then cases3 = "true" else cases3 = "false" end local detected = false detected = detected or true]] return mw.title.getCurrentTitle() end function p.ifexist() return tonumber(frame:expandTemplate{ title = "User:Helpme2222/Sandbox", args = {frame.args[1]} }) end function p.performance() for i = 1,5000000 do p.tableMake() end return end function p.performance1() local var = "" for i = 1,100000000 do if var[1] then end end return end function p.performance2() local var = "" for i = 1,100000000 do if type(var) == "table" then end end return end p.counter = 0 function p.tableBake() p.counter = p.counter + 1 return p.counter end function p.tableMake() p.counter = 0 local var = { keyword = {p.tableBake()}, kljuczslovo = {p.tableBake()}, yaoshiwenzi = {p.tableBake()}, llavepalabra = {p.tableBake()}, } return var end function p.expand() return frame:expandTemplate{ title = "User:Helpme2222/Sandbox" } end function p.title3() return mw.title.getCurrentTitle().fullText end function p.getBallAttribute() return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4]) end function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration ) if _ then return mw.text.nowiki(ballData) else return "" end end function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att] if ballData[1] then return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)] end return ballData end function p.emojiCodepointInternal( emoji ) emoji = mw.text.trim( emoji ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.emojiCodepoint() local emoji = mw.text.trim( mw.getCurrentFrame().args[1] ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.title2() local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA", ["Module:Sandbox/22"] = "Module:Sandbox/22" } } local concepts = { grammatical_cases, } local PAGENAME = tostring(mw.title.getCurrentTitle()) -- local detected = false local out = {} local debugger = {} for i, group in ipairs(concepts) do if group.inessive[PAGENAME] then table.insert(debugger,"Okay, managed TRUE on "..i..":"..group.ablative) -- detected = detected or true local formattedGroup = {} for memberKey, memberRendervalue in pairs(group.inessive) do table.insert(formattedGroup, "<li>" .. memberRendervalue .. "</li>") end table.insert(out, "<h3>" .. group.ablative .. "</h3><ul>" .. table.concat( formattedGroup ) .. "</ul>" ) table.insert(out, "this sucks ass") return "this sucks ass" else table.insert(debugger,"Managed FALSE on "..i..":"..group.ablative..". Comparator: "..PAGENAME..", ") end end return debugger --[[ if detected then return tostring(detected) .. table.concat(out) end return --]] end function p.sanitizeChar(capture) -- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited if capture == "?" then return "" elseif capture == "#" then return "No. " end end function p.sanitizeTitle() local title = frame.args[1] return mw.ustring.gsub( title, "(?:\\?|\\#)", p.sanitizeChar) end function p.sanitizeTitletest2() return mw.ustring.gsub( "Ki vagy? (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)", "(?:\\?|\\#)", p.sanitizeChar()) end function p.safetyTest() if string.len(frame.args[1]) < 500 then return frame.args[1] else return "" end end --[[ function array_iter(t) local i = 0 return function () i = i + 1 return t[i] end end --]] function p.listInline() --[[ This function is a godsend! https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]] local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return mw.text.listToText( spritedTable , ", ", ", and " ) end function p.listNavbox() -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return table.concat( spritedTable , " • ") end function p.listCountInternal(table) local count = 0 for index,value in ipairs( table ) do count = count + 1 end return count end function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn function p.tableTest() return balls[1][1] end function p.todaysBall() --[[ https://www.lua.org/pil/3.6.html The use of explicit indexing here is not strictly semantic; it's just to emphasize the rotation order. ]] local unixDay = math.floor(os.time()/86400) local ballTotalCount = 0 local ballCounts = {} for whichTable,subtable in ipairs(balls) do ballCounts[whichTable] = #subtable ballTotalCount = ballTotalCount + ballCounts[whichTable] end local cycleDayIndex, cycleStartingBall = unixDay % ballTotalCount, math.floor(unixDay/ballTotalCount) % ballTotalCount local cycleTodaysBall if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end --[[ debug return cycleTodaysBall .. " " .. ballTotalCount .. " " .. os.time() .. " " .. math.floor(os.time()/86400) .. " " .. cycleDayIndex .. " " .. cycleStartingBall ]] for whichTable = 1, #balls do if ballCounts[whichTable] > cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end end --]] end return p; 4ytbjqmymux5vl7wq8zsev8pwolhnii 2834931 2834930 2026-09-28T23:42:05Z Helpme2222 3106525 2834931 Scribunto text/plain -- My function for teaching grammar local p = {}; local frame = mw.getCurrentFrame() local grammatical_moods = { "Jelen ido", "Mult ido", "Felszolito mod", "Felteteles mod" } local balls = { grammatical_moods } local toggleButton = mw.html.create( 'span' ) :css({ float = "right", ["margin-right"] = "0", ["font-weight"] = "normal" }) :addClass("mw-customtoggle-Infobox") :node("[Click to toggle]") function p.subheading( discrim, b_content, content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node((discrim) and b_content or content) :node((discrim) and toggleButton) ) end function p.s_subheading( s_content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible mw-collapsed" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node(s_content) :node(toggleButton) ) end function p.row( discrim, name, data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(name) ) :node( mw.html.create( 'td' ) :node(data) ) end function p.s_row( s_name, s_data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass("mw-collapsible mw-collapsed") :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(s_name) ) :node( mw.html.create( 'td' ) :node(s_data) ) end function p.var(symbol) return frame:expandTemplate{ title = "Template:Var", args = {symbol} } end function p.iseq() return table.concat({ frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/'") and " is equal to User:Helpme2222/Sandbox/'" or " is not equal to User:Helpme2222/Sandbox/'"), frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/&#039;") and " is equal to User:Helpme2222/Sandbox/&#039;" or " is not equal to User:Helpme2222/Sandbox/&#039;"), frame.args[1] .. ( (frame.args[1] == "Helpme2222/Sandbox/'") and " is equal to Helpme2222/Sandbox/'" or " is not equal to Helpme2222/Sandbox/'"), frame.args[1] .. ( (frame.args[1] == "Helpme2222/Sandbox/&#039;") and " is equal to Helpme2222/Sandbox/&#039;" or " is not equal to Helpme2222/Sandbox/&#039;"), }) end function p.infoboxWBB() -- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html --[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node Thankfully, these operations interpret passed nils as no-ops (I'm technically using false though. Somehow that works anyways?)]] local frame = mw.getCurrentFrame() local ballname = frame.args["Title"] or tostring(mw.title.getCurrentTitle()) local Entities = p -- require("Module:Entities") local featureCount = tonumber(frame.args["Feature count"]) or 1 local s_featureCount = tonumber(frame.args["SUPER Feature count"]) or featureCount local matchCount = { total = mw.site.stats.pagesInCategory( ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local matchWinCount = { standard = mw.site.stats.pagesInCategory( ballname.."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local s_matchCount = { total = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local s_matchWinCount = { standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local isSuper = frame.args["SUPER Feature count"] or frame.args["SUPER Type"] or frame.args["SUPER Debuted"] or (s_matchCount.total > 0) or nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra "or nil" had to be added since this is used in :addClass calls. -- TITLE local infobox = mw.html.create( 'table' ) :addClass("wikitable") :css({ float = "right"; ["margin-left"] = "10px" }) :node( mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass((isSuper) and "mw-collapsible" or nil) :node("<big>" .. ballname .. "</big>") ) ) :node((isSuper) and mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass("mw-collapsible mw-collapsed") :node("<big>" .. (frame.args["SUPER Title"] or ("Super " .. ballname)) .. "</big>") ) ) -- SECTION: Identification -- Section Heading :node( p.subheading( (frame.args["SUPER Type"]), "Base Identification", "Basic Info" ) ) :node( (frame.args["SUPER Type"]) and p.s_subheading( "Super Identification" ) ) -- Type :node( p.row( (frame.args["SUPER Type"]), "Type", frame.args["Type"] ) ) :node( (frame.args["SUPER Type"]) and p.s_row( "Type", frame.args["SUPER Type"] ) ) -- Color local color = frame.args["Color"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "color" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Color") ) :node( mw.html.create( 'td' ) :node( (color ~= "") and frame:expandTemplate{ title = "Color", args = {color} } or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add color...</i>]]" ) ) ) -- Emoji local emoji = frame.args["Emoji"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "emoji" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Emoji") ) :node( mw.html.create( 'td' ) :node( (emoji ~= "") and ( emoji .. " <code>" .. p.emojiCodepointInternal(emoji) .. "</code>") or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add emoji...</i>]]" ) ) ) -- SECTION: Scaling Characteristics --[[ Some data will be collected first. ]] -- (Base) Behavior local features = {} local starts = {} local scalings = {} if (featureCount > 1) then for i = 1, featureCount do table.insert(features, { frame.args["Feature "..i], " (" .. p.var(frame.args["Var "..i]) .. ")" }) table.insert(starts, { p.var( frame.args["Var0 "..i] or (frame.args["Var "..i].."₀") ) .. ": ", frame.args["Starting value "..i] }) table.insert(scalings, { p.var( "Δ"..(frame.args["Var "..i]) ) .. ": ", frame.args["Scaling "..i] }) end else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1 features[1] = {frame.args["Feature"],""} starts[1] = {"",frame.args["Starting value"]} scalings[1] = {"",frame.args["Scaling"]} end -- Super Behavior local s_features local s_starts local s_scalings -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default. local has_s_features; local has_s_starts; local has_s_scalings; if (frame.args["SUPER Feature count"]) then if s_featureCount > 1 then -- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data s_features = mw.clone(features) s_starts = mw.clone(starts) s_scalings = mw.clone(scalings) -- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors. if s_featureCount > featureCount then for i = featureCount + 1, s_featureCount do table.insert( s_features, {"",""} ) table.insert( s_starts, {"",""} ) table.insert( s_scalings, {"",""} ) end elseif s_featureCount < featureCount then for i = s_featureCount + 1, featureCount do s_features[i] = nil s_starts[i] = nil s_scalings[i] = nil end end -- Parse super data, set the has_s_... flags if triggered for i = 1,s_featureCount do if frame.args["SUPER Feature "..i] then s_features[i][1] = frame.args["SUPER Feature "..i] if (not has_s_features) then has_s_features = true end end if frame.args["SUPER Starting value "..i] then s_starts[i][2] = frame.args["SUPER Starting value "..i] if (not has_s_starts) then has_s_starts = true end end if frame.args["SUPER Scaling "..i] then s_scalings[i][2] = frame.args["SUPER Scaling "..i] if (not has_s_scalings) then has_s_scalings = true end end if frame.args["SUPER Var "..i] then s_features[i][2] = " (" .. p.var(frame.args["SUPER Var "..i]) .. ")" s_starts[i][1] = p.var( frame.args["SUPER Var0 "..i] or (frame.args["SUPER Var "..i].."₀") ) .. ": " s_scalings[i][1] = p.var( '<span style="font-style:normal">Δ</span>'..(frame.args["SUPER Var "..i]) ) .. ": " if (not has_s_features) then has_s_features = true end if (not has_s_starts) then has_s_starts = true end if (not has_s_scalings) then has_s_scalings = true end end s_features[i] = table.concat(s_features[i]) s_starts[i] = table.concat(s_starts[i]) s_scalings[i] = table.concat(s_scalings[i]) end s_features = table.concat(s_features,"<br>") s_starts = table.concat(s_starts,"<br>") s_scalings = table.concat(s_scalings,"<br>") else -- # of super features = 1? just write directly to the variable if frame.args["SUPER Var"] or frame.args["SUPER Feature"] then has_s_features = true s_features = frame.args["SUPER Feature"] end if frame.args["SUPER Var0"] or frame.args["SUPER Var"] or frame.args["SUPER Starting value"] then has_s_starts = true s_starts = frame.args["SUPER Starting value"] end if frame.args["SUPER Scaling"] then has_s_scalings = true s_scalings = frame.args["SUPER Scaling"] end end end -- finalize base data as strings now that work is otherwise complete if (featureCount > 1) then for i = 1, featureCount do features[i] = table.concat(features[i]) starts[i] = table.concat(starts[i]) scalings[i] = table.concat(scalings[i]) end features = table.concat(features,"<br>") starts = table.concat(starts,"<br>") scalings = table.concat(scalings,"<br>") else features = features[1][1] starts = starts[1][2] scalings = scalings[1][2] end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (frame.args["SUPER Feature count"]), "Base Scaling Characteristics", "Scaling Characteristics" ) ) :node( (frame.args["SUPER Feature count"]) and p.s_subheading( "Super Scaling Characteristics" ) ) -- Features :node( p.row( (has_s_features), (featureCount > 1) and "Features" or ( (frame.args["Var"]) and ("Feature (" .. p.var(frame.args["Var"]) .. ")") or "Feature" ), features ) ) :node( (has_s_features) and p.s_row( (s_featureCount > 1) and "Features" or ( (frame.args["SUPER Var"]) and ("Feature (" .. p.var(frame.args["SUPER Var"]) .. ")") or "Feature" ), s_features ) ) -- Starting values :node( p.row( (has_s_starts), (featureCount > 1) and "Starting values" or ( (frame.args["Var"]) and ("Starting value (" .. ( p.var(frame.args["Var0"] or (frame.args["Var"].."₀") ) ) .. ")") or "Starting value" ), starts ) ) :node( (has_s_starts) and p.s_row( (s_featureCount > 1) and "Starting values" or ( (frame.args["SUPER Var"]) and ("Starting value (" .. ( p.var(frame.args["SUPER Var0"] or (frame.args["SUPER Var"].."₀") ) ) .. ")") or "Starting value" ), s_starts ) ) -- Scaling :node( p.row( (has_s_scalings), "Scaling", scalings) ) :node( (has_s_scalings) and p.s_row("Scaling", s_scalings) ) -- SECTION: Career --[[ Some data will be collected first. ]] -- (Base) Games played local gamesPlayedText = {} if matchCount.total > 0 then table.insert(gamesPlayedText, matchCount.total .. " (total)") if matchCount.standard > 0 then table.insert(gamesPlayedText, matchCount.standard .. " (standard)") end end -- (Base) Winrate local winrateText = {} if matchCount.total > 0 then if matchCount.standard > 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. "% (standard)" ) end end -- Super Games played local s_gamesPlayedText = {} if s_matchCount.total > 0 then table.insert(s_gamesPlayedText, s_matchCount.total .. " (total)") if s_matchCount.standard > 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. " (standard)") end end -- Super Winrate local s_winrateText = {} if s_matchCount.total > 0 then if s_matchCount.standard > 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. "% (standard)" ) end end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (isSuper), "Base Career", "Career ") ) :node( (isSuper) and p.s_subheading( "Super Career") ) -- Debuted :node( p.row( (isSuper), "Debut", frame.args["Debut"] ) ) :node( (isSuper) and p.s_row("Debut", frame.args["SUPER Debut"]) ) -- Games played :node( p.row( (isSuper), "Games played", (#gamesPlayedText > 0) and table.concat(gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Games played", (#s_gamesPlayedText > 0) and table.concat(s_gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) -- Winrate :node( p.row( (isSuper), "Winrate", (#winrateText > 0) and table.concat(winrateText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Winrate", (#s_winrateText > 0) and table.concat(s_winrateText, "<br>") or "<i>Not enough data...</i>" ) ) return tostring(infobox) end function p.referenceTest() local element = mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible" ) :css( "text-align","left" ) local bigElement = mw.html.create( "div" ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface") ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface2") ) return tostring(element) .. "\n" .. tostring(bigElement) .. "\n" end function p.roundedPct(num, den) return (num/den % 0.01 < 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100) end function p.title() --[[local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA" } } local cases2 = "ba" local cases3 = "ba" -- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = "true" else cases2 = "false" end -- if "ANCOVA" == frame:getParent():getTitle() then cases3 = "true" else cases3 = "false" end local detected = false detected = detected or true]] return mw.title.getCurrentTitle() end function p.ifexist() return tonumber(frame:expandTemplate{ title = "User:Helpme2222/Sandbox", args = {frame.args[1]} }) end function p.performance() for i = 1,5000000 do p.tableMake() end return end function p.performance1() local var = "" for i = 1,100000000 do if var[1] then end end return end function p.performance2() local var = "" for i = 1,100000000 do if type(var) == "table" then end end return end p.counter = 0 function p.tableBake() p.counter = p.counter + 1 return p.counter end function p.tableMake() p.counter = 0 local var = { keyword = {p.tableBake()}, kljuczslovo = {p.tableBake()}, yaoshiwenzi = {p.tableBake()}, llavepalabra = {p.tableBake()}, } return var end function p.expand() return frame:expandTemplate{ title = "User:Helpme2222/Sandbox" } end function p.title3() return mw.title.getCurrentTitle().fullText end function p.getBallAttribute() return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4]) end function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration ) if _ then return mw.text.nowiki(ballData) else return "" end end function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att] if ballData[1] then return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)] end return ballData end function p.emojiCodepointInternal( emoji ) emoji = mw.text.trim( emoji ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.emojiCodepoint() local emoji = mw.text.trim( mw.getCurrentFrame().args[1] ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.title2() local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA", ["Module:Sandbox/22"] = "Module:Sandbox/22" } } local concepts = { grammatical_cases, } local PAGENAME = tostring(mw.title.getCurrentTitle()) -- local detected = false local out = {} local debugger = {} for i, group in ipairs(concepts) do if group.inessive[PAGENAME] then table.insert(debugger,"Okay, managed TRUE on "..i..":"..group.ablative) -- detected = detected or true local formattedGroup = {} for memberKey, memberRendervalue in pairs(group.inessive) do table.insert(formattedGroup, "<li>" .. memberRendervalue .. "</li>") end table.insert(out, "<h3>" .. group.ablative .. "</h3><ul>" .. table.concat( formattedGroup ) .. "</ul>" ) table.insert(out, "this sucks ass") return "this sucks ass" else table.insert(debugger,"Managed FALSE on "..i..":"..group.ablative..". Comparator: "..PAGENAME..", ") end end return debugger --[[ if detected then return tostring(detected) .. table.concat(out) end return --]] end function p.sanitizeChar(capture) -- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited if capture == "?" then return "" elseif capture == "#" then return "No. " end end function p.sanitizeTitle() local title = frame.args[1] return mw.ustring.gsub( title, "(?:\\?|\\#)", p.sanitizeChar) end function p.sanitizeTitletest2() return mw.ustring.gsub( "Ki vagy? (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)", "(?:\\?|\\#)", p.sanitizeChar()) end function p.safetyTest() if string.len(frame.args[1]) < 500 then return frame.args[1] else return "" end end --[[ function array_iter(t) local i = 0 return function () i = i + 1 return t[i] end end --]] function p.listInline() --[[ This function is a godsend! https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]] local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return mw.text.listToText( spritedTable , ", ", ", and " ) end function p.listNavbox() -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return table.concat( spritedTable , " • ") end function p.listCountInternal(table) local count = 0 for index,value in ipairs( table ) do count = count + 1 end return count end function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn function p.tableTest() return balls[1][1] end function p.todaysBall() --[[ https://www.lua.org/pil/3.6.html The use of explicit indexing here is not strictly semantic; it's just to emphasize the rotation order. ]] local unixDay = math.floor(os.time()/86400) local ballTotalCount = 0 local ballCounts = {} for whichTable,subtable in ipairs(balls) do ballCounts[whichTable] = #subtable ballTotalCount = ballTotalCount + ballCounts[whichTable] end local cycleDayIndex, cycleStartingBall = unixDay % ballTotalCount, math.floor(unixDay/ballTotalCount) % ballTotalCount local cycleTodaysBall if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end --[[ debug return cycleTodaysBall .. " " .. ballTotalCount .. " " .. os.time() .. " " .. math.floor(os.time()/86400) .. " " .. cycleDayIndex .. " " .. cycleStartingBall ]] for whichTable = 1, #balls do if ballCounts[whichTable] > cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end end --]] end return p; 10vm9kix8gex51ndn5039fvybzmzneq 2834932 2834931 2026-09-28T23:42:36Z Helpme2222 3106525 2834932 Scribunto text/plain -- My function for teaching grammar local p = {}; local frame = mw.getCurrentFrame() local grammatical_moods = { "Jelen ido", "Mult ido", "Felszolito mod", "Felteteles mod" } local balls = { grammatical_moods } local toggleButton = mw.html.create( 'span' ) :css({ float = "right", ["margin-right"] = "0", ["font-weight"] = "normal" }) :addClass("mw-customtoggle-Infobox") :node("[Click to toggle]") function p.subheading( discrim, b_content, content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node((discrim) and b_content or content) :node((discrim) and toggleButton) ) end function p.s_subheading( s_content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible mw-collapsed" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node(s_content) :node(toggleButton) ) end function p.row( discrim, name, data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(name) ) :node( mw.html.create( 'td' ) :node(data) ) end function p.s_row( s_name, s_data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass("mw-collapsible mw-collapsed") :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(s_name) ) :node( mw.html.create( 'td' ) :node(s_data) ) end function p.var(symbol) return frame:expandTemplate{ title = "Template:Var", args = {symbol} } end function p.iseq() return table.concat({ frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/'") and " is equal to User:Helpme2222/Sandbox/'" or " is not equal to User:Helpme2222/Sandbox/'"), frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/&#039;") and " is equal to User:Helpme2222/Sandbox/&#039;" or " is not equal to User:Helpme2222/Sandbox/&#039;"), frame.args[1] .. ( (frame.args[1] == "Helpme2222/Sandbox/'") and " is equal to Helpme2222/Sandbox/'" or " is not equal to Helpme2222/Sandbox/'"), frame.args[1] .. ( (frame.args[1] == "Helpme2222/Sandbox/&#039;") and " is equal to Helpme2222/Sandbox/&#039;" or " is not equal to Helpme2222/Sandbox/&#039;"), }, "<br>") end function p.infoboxWBB() -- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html --[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node Thankfully, these operations interpret passed nils as no-ops (I'm technically using false though. Somehow that works anyways?)]] local frame = mw.getCurrentFrame() local ballname = frame.args["Title"] or tostring(mw.title.getCurrentTitle()) local Entities = p -- require("Module:Entities") local featureCount = tonumber(frame.args["Feature count"]) or 1 local s_featureCount = tonumber(frame.args["SUPER Feature count"]) or featureCount local matchCount = { total = mw.site.stats.pagesInCategory( ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local matchWinCount = { standard = mw.site.stats.pagesInCategory( ballname.."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local s_matchCount = { total = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local s_matchWinCount = { standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local isSuper = frame.args["SUPER Feature count"] or frame.args["SUPER Type"] or frame.args["SUPER Debuted"] or (s_matchCount.total > 0) or nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra "or nil" had to be added since this is used in :addClass calls. -- TITLE local infobox = mw.html.create( 'table' ) :addClass("wikitable") :css({ float = "right"; ["margin-left"] = "10px" }) :node( mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass((isSuper) and "mw-collapsible" or nil) :node("<big>" .. ballname .. "</big>") ) ) :node((isSuper) and mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass("mw-collapsible mw-collapsed") :node("<big>" .. (frame.args["SUPER Title"] or ("Super " .. ballname)) .. "</big>") ) ) -- SECTION: Identification -- Section Heading :node( p.subheading( (frame.args["SUPER Type"]), "Base Identification", "Basic Info" ) ) :node( (frame.args["SUPER Type"]) and p.s_subheading( "Super Identification" ) ) -- Type :node( p.row( (frame.args["SUPER Type"]), "Type", frame.args["Type"] ) ) :node( (frame.args["SUPER Type"]) and p.s_row( "Type", frame.args["SUPER Type"] ) ) -- Color local color = frame.args["Color"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "color" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Color") ) :node( mw.html.create( 'td' ) :node( (color ~= "") and frame:expandTemplate{ title = "Color", args = {color} } or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add color...</i>]]" ) ) ) -- Emoji local emoji = frame.args["Emoji"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "emoji" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Emoji") ) :node( mw.html.create( 'td' ) :node( (emoji ~= "") and ( emoji .. " <code>" .. p.emojiCodepointInternal(emoji) .. "</code>") or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add emoji...</i>]]" ) ) ) -- SECTION: Scaling Characteristics --[[ Some data will be collected first. ]] -- (Base) Behavior local features = {} local starts = {} local scalings = {} if (featureCount > 1) then for i = 1, featureCount do table.insert(features, { frame.args["Feature "..i], " (" .. p.var(frame.args["Var "..i]) .. ")" }) table.insert(starts, { p.var( frame.args["Var0 "..i] or (frame.args["Var "..i].."₀") ) .. ": ", frame.args["Starting value "..i] }) table.insert(scalings, { p.var( "Δ"..(frame.args["Var "..i]) ) .. ": ", frame.args["Scaling "..i] }) end else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1 features[1] = {frame.args["Feature"],""} starts[1] = {"",frame.args["Starting value"]} scalings[1] = {"",frame.args["Scaling"]} end -- Super Behavior local s_features local s_starts local s_scalings -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default. local has_s_features; local has_s_starts; local has_s_scalings; if (frame.args["SUPER Feature count"]) then if s_featureCount > 1 then -- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data s_features = mw.clone(features) s_starts = mw.clone(starts) s_scalings = mw.clone(scalings) -- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors. if s_featureCount > featureCount then for i = featureCount + 1, s_featureCount do table.insert( s_features, {"",""} ) table.insert( s_starts, {"",""} ) table.insert( s_scalings, {"",""} ) end elseif s_featureCount < featureCount then for i = s_featureCount + 1, featureCount do s_features[i] = nil s_starts[i] = nil s_scalings[i] = nil end end -- Parse super data, set the has_s_... flags if triggered for i = 1,s_featureCount do if frame.args["SUPER Feature "..i] then s_features[i][1] = frame.args["SUPER Feature "..i] if (not has_s_features) then has_s_features = true end end if frame.args["SUPER Starting value "..i] then s_starts[i][2] = frame.args["SUPER Starting value "..i] if (not has_s_starts) then has_s_starts = true end end if frame.args["SUPER Scaling "..i] then s_scalings[i][2] = frame.args["SUPER Scaling "..i] if (not has_s_scalings) then has_s_scalings = true end end if frame.args["SUPER Var "..i] then s_features[i][2] = " (" .. p.var(frame.args["SUPER Var "..i]) .. ")" s_starts[i][1] = p.var( frame.args["SUPER Var0 "..i] or (frame.args["SUPER Var "..i].."₀") ) .. ": " s_scalings[i][1] = p.var( '<span style="font-style:normal">Δ</span>'..(frame.args["SUPER Var "..i]) ) .. ": " if (not has_s_features) then has_s_features = true end if (not has_s_starts) then has_s_starts = true end if (not has_s_scalings) then has_s_scalings = true end end s_features[i] = table.concat(s_features[i]) s_starts[i] = table.concat(s_starts[i]) s_scalings[i] = table.concat(s_scalings[i]) end s_features = table.concat(s_features,"<br>") s_starts = table.concat(s_starts,"<br>") s_scalings = table.concat(s_scalings,"<br>") else -- # of super features = 1? just write directly to the variable if frame.args["SUPER Var"] or frame.args["SUPER Feature"] then has_s_features = true s_features = frame.args["SUPER Feature"] end if frame.args["SUPER Var0"] or frame.args["SUPER Var"] or frame.args["SUPER Starting value"] then has_s_starts = true s_starts = frame.args["SUPER Starting value"] end if frame.args["SUPER Scaling"] then has_s_scalings = true s_scalings = frame.args["SUPER Scaling"] end end end -- finalize base data as strings now that work is otherwise complete if (featureCount > 1) then for i = 1, featureCount do features[i] = table.concat(features[i]) starts[i] = table.concat(starts[i]) scalings[i] = table.concat(scalings[i]) end features = table.concat(features,"<br>") starts = table.concat(starts,"<br>") scalings = table.concat(scalings,"<br>") else features = features[1][1] starts = starts[1][2] scalings = scalings[1][2] end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (frame.args["SUPER Feature count"]), "Base Scaling Characteristics", "Scaling Characteristics" ) ) :node( (frame.args["SUPER Feature count"]) and p.s_subheading( "Super Scaling Characteristics" ) ) -- Features :node( p.row( (has_s_features), (featureCount > 1) and "Features" or ( (frame.args["Var"]) and ("Feature (" .. p.var(frame.args["Var"]) .. ")") or "Feature" ), features ) ) :node( (has_s_features) and p.s_row( (s_featureCount > 1) and "Features" or ( (frame.args["SUPER Var"]) and ("Feature (" .. p.var(frame.args["SUPER Var"]) .. ")") or "Feature" ), s_features ) ) -- Starting values :node( p.row( (has_s_starts), (featureCount > 1) and "Starting values" or ( (frame.args["Var"]) and ("Starting value (" .. ( p.var(frame.args["Var0"] or (frame.args["Var"].."₀") ) ) .. ")") or "Starting value" ), starts ) ) :node( (has_s_starts) and p.s_row( (s_featureCount > 1) and "Starting values" or ( (frame.args["SUPER Var"]) and ("Starting value (" .. ( p.var(frame.args["SUPER Var0"] or (frame.args["SUPER Var"].."₀") ) ) .. ")") or "Starting value" ), s_starts ) ) -- Scaling :node( p.row( (has_s_scalings), "Scaling", scalings) ) :node( (has_s_scalings) and p.s_row("Scaling", s_scalings) ) -- SECTION: Career --[[ Some data will be collected first. ]] -- (Base) Games played local gamesPlayedText = {} if matchCount.total > 0 then table.insert(gamesPlayedText, matchCount.total .. " (total)") if matchCount.standard > 0 then table.insert(gamesPlayedText, matchCount.standard .. " (standard)") end end -- (Base) Winrate local winrateText = {} if matchCount.total > 0 then if matchCount.standard > 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. "% (standard)" ) end end -- Super Games played local s_gamesPlayedText = {} if s_matchCount.total > 0 then table.insert(s_gamesPlayedText, s_matchCount.total .. " (total)") if s_matchCount.standard > 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. " (standard)") end end -- Super Winrate local s_winrateText = {} if s_matchCount.total > 0 then if s_matchCount.standard > 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. "% (standard)" ) end end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (isSuper), "Base Career", "Career ") ) :node( (isSuper) and p.s_subheading( "Super Career") ) -- Debuted :node( p.row( (isSuper), "Debut", frame.args["Debut"] ) ) :node( (isSuper) and p.s_row("Debut", frame.args["SUPER Debut"]) ) -- Games played :node( p.row( (isSuper), "Games played", (#gamesPlayedText > 0) and table.concat(gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Games played", (#s_gamesPlayedText > 0) and table.concat(s_gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) -- Winrate :node( p.row( (isSuper), "Winrate", (#winrateText > 0) and table.concat(winrateText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Winrate", (#s_winrateText > 0) and table.concat(s_winrateText, "<br>") or "<i>Not enough data...</i>" ) ) return tostring(infobox) end function p.referenceTest() local element = mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible" ) :css( "text-align","left" ) local bigElement = mw.html.create( "div" ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface") ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface2") ) return tostring(element) .. "\n" .. tostring(bigElement) .. "\n" end function p.roundedPct(num, den) return (num/den % 0.01 < 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100) end function p.title() --[[local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA" } } local cases2 = "ba" local cases3 = "ba" -- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = "true" else cases2 = "false" end -- if "ANCOVA" == frame:getParent():getTitle() then cases3 = "true" else cases3 = "false" end local detected = false detected = detected or true]] return mw.title.getCurrentTitle() end function p.ifexist() return tonumber(frame:expandTemplate{ title = "User:Helpme2222/Sandbox", args = {frame.args[1]} }) end function p.performance() for i = 1,5000000 do p.tableMake() end return end function p.performance1() local var = "" for i = 1,100000000 do if var[1] then end end return end function p.performance2() local var = "" for i = 1,100000000 do if type(var) == "table" then end end return end p.counter = 0 function p.tableBake() p.counter = p.counter + 1 return p.counter end function p.tableMake() p.counter = 0 local var = { keyword = {p.tableBake()}, kljuczslovo = {p.tableBake()}, yaoshiwenzi = {p.tableBake()}, llavepalabra = {p.tableBake()}, } return var end function p.expand() return frame:expandTemplate{ title = "User:Helpme2222/Sandbox" } end function p.title3() return mw.title.getCurrentTitle().fullText end function p.getBallAttribute() return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4]) end function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration ) if _ then return mw.text.nowiki(ballData) else return "" end end function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att] if ballData[1] then return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)] end return ballData end function p.emojiCodepointInternal( emoji ) emoji = mw.text.trim( emoji ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.emojiCodepoint() local emoji = mw.text.trim( mw.getCurrentFrame().args[1] ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.title2() local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA", ["Module:Sandbox/22"] = "Module:Sandbox/22" } } local concepts = { grammatical_cases, } local PAGENAME = tostring(mw.title.getCurrentTitle()) -- local detected = false local out = {} local debugger = {} for i, group in ipairs(concepts) do if group.inessive[PAGENAME] then table.insert(debugger,"Okay, managed TRUE on "..i..":"..group.ablative) -- detected = detected or true local formattedGroup = {} for memberKey, memberRendervalue in pairs(group.inessive) do table.insert(formattedGroup, "<li>" .. memberRendervalue .. "</li>") end table.insert(out, "<h3>" .. group.ablative .. "</h3><ul>" .. table.concat( formattedGroup ) .. "</ul>" ) table.insert(out, "this sucks ass") return "this sucks ass" else table.insert(debugger,"Managed FALSE on "..i..":"..group.ablative..". Comparator: "..PAGENAME..", ") end end return debugger --[[ if detected then return tostring(detected) .. table.concat(out) end return --]] end function p.sanitizeChar(capture) -- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited if capture == "?" then return "" elseif capture == "#" then return "No. " end end function p.sanitizeTitle() local title = frame.args[1] return mw.ustring.gsub( title, "(?:\\?|\\#)", p.sanitizeChar) end function p.sanitizeTitletest2() return mw.ustring.gsub( "Ki vagy? (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)", "(?:\\?|\\#)", p.sanitizeChar()) end function p.safetyTest() if string.len(frame.args[1]) < 500 then return frame.args[1] else return "" end end --[[ function array_iter(t) local i = 0 return function () i = i + 1 return t[i] end end --]] function p.listInline() --[[ This function is a godsend! https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]] local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return mw.text.listToText( spritedTable , ", ", ", and " ) end function p.listNavbox() -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return table.concat( spritedTable , " • ") end function p.listCountInternal(table) local count = 0 for index,value in ipairs( table ) do count = count + 1 end return count end function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn function p.tableTest() return balls[1][1] end function p.todaysBall() --[[ https://www.lua.org/pil/3.6.html The use of explicit indexing here is not strictly semantic; it's just to emphasize the rotation order. ]] local unixDay = math.floor(os.time()/86400) local ballTotalCount = 0 local ballCounts = {} for whichTable,subtable in ipairs(balls) do ballCounts[whichTable] = #subtable ballTotalCount = ballTotalCount + ballCounts[whichTable] end local cycleDayIndex, cycleStartingBall = unixDay % ballTotalCount, math.floor(unixDay/ballTotalCount) % ballTotalCount local cycleTodaysBall if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end --[[ debug return cycleTodaysBall .. " " .. ballTotalCount .. " " .. os.time() .. " " .. math.floor(os.time()/86400) .. " " .. cycleDayIndex .. " " .. cycleStartingBall ]] for whichTable = 1, #balls do if ballCounts[whichTable] > cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end end --]] end return p; aiq379h8aqxji6xt89haq55g0x21fnc 2834933 2834932 2026-09-28T23:46:34Z Helpme2222 3106525 2834933 Scribunto text/plain -- My function for teaching grammar local p = {}; local frame = mw.getCurrentFrame() local grammatical_moods = { "Jelen ido", "Mult ido", "Felszolito mod", "Felteteles mod" } local balls = { grammatical_moods } local toggleButton = mw.html.create( 'span' ) :css({ float = "right", ["margin-right"] = "0", ["font-weight"] = "normal" }) :addClass("mw-customtoggle-Infobox") :node("[Click to toggle]") function p.subheading( discrim, b_content, content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node((discrim) and b_content or content) :node((discrim) and toggleButton) ) end function p.s_subheading( s_content ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible mw-collapsed" ) :node( mw.html.create( 'th' ) :attr( "colspan", "2" ) :css( "text-align","left" ) :node(s_content) :node(toggleButton) ) end function p.row( discrim, name, data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( (discrim) and "mw-collapsible" ) :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(name) ) :node( mw.html.create( 'td' ) :node(data) ) end function p.s_row( s_name, s_data ) return mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass("mw-collapsible mw-collapsed") :node( mw.html.create( 'td' ) :css( "font-weight","bold" ) :node(s_name) ) :node( mw.html.create( 'td' ) :node(s_data) ) end function p.var(symbol) return frame:expandTemplate{ title = "Template:Var", args = {symbol} } end function p.iseq() return table.concat({ frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/'") and " is equal to User:Helpme2222/Sandbox/'" or " is not equal to User:Helpme2222/Sandbox/'"), frame.args[1] .. ( (frame.args[1] == "User:Helpme2222/Sandbox/&#039;") and " is equal to User:Helpme2222/Sandbox/&#039;" or " is not equal to User:Helpme2222/Sandbox/&#039;"), frame.args[1] .. ( (frame.args[1] == "Helpme2222/Sandbox/'") and " is equal to Helpme2222/Sandbox/'" or " is not equal to Helpme2222/Sandbox/'"), frame.args[1] .. ( (frame.args[1] == "Helpme2222/Sandbox/&#039;") and " is equal to Helpme2222/Sandbox/&#039;" or " is not equal to Helpme2222/Sandbox/&#039;"), }, "<br>") end function p.iseqNamed() return table.concat({ frame.args.arg .. ( (frame.args.arg == "User:Helpme2222/Sandbox/'") and " is equal to User:Helpme2222/Sandbox/'" or " is not equal to User:Helpme2222/Sandbox/'"), frame.args.arg .. ( (frame.args.arg == "User:Helpme2222/Sandbox/&#039;") and " is equal to User:Helpme2222/Sandbox/&#039;" or " is not equal to User:Helpme2222/Sandbox/&#039;"), frame.args.arg .. ( (frame.args.arg == "Helpme2222/Sandbox/'") and " is equal to Helpme2222/Sandbox/'" or " is not equal to Helpme2222/Sandbox/'"), frame.args.arg .. ( (frame.args.arg == "Helpme2222/Sandbox/&#039;") and " is equal to Helpme2222/Sandbox/&#039;" or " is not equal to Helpme2222/Sandbox/&#039;"), }, "<br>") end function p.infoboxWBB() -- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html --[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node Thankfully, these operations interpret passed nils as no-ops (I'm technically using false though. Somehow that works anyways?)]] local frame = mw.getCurrentFrame() local ballname = frame.args["Title"] or tostring(mw.title.getCurrentTitle()) local Entities = p -- require("Module:Entities") local featureCount = tonumber(frame.args["Feature count"]) or 1 local s_featureCount = tonumber(frame.args["SUPER Feature count"]) or featureCount local matchCount = { total = mw.site.stats.pagesInCategory( ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local matchWinCount = { standard = mw.site.stats.pagesInCategory( ballname.."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local s_matchCount = { total = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History", "pages" ), standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard", "pages" ), } local s_matchWinCount = { standard = mw.site.stats.pagesInCategory( "Super " .. ballname .."'s Match History/Weapon Ball Battles/Standard/Won", "pages" ), } local isSuper = frame.args["SUPER Feature count"] or frame.args["SUPER Type"] or frame.args["SUPER Debuted"] or (s_matchCount.total > 0) or nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra "or nil" had to be added since this is used in :addClass calls. -- TITLE local infobox = mw.html.create( 'table' ) :addClass("wikitable") :css({ float = "right"; ["margin-left"] = "10px" }) :node( mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass((isSuper) and "mw-collapsible" or nil) :node("<big>" .. ballname .. "</big>") ) ) :node((isSuper) and mw.html.create( 'tr' ) :node( mw.html.create( 'th' ) :attr({ colspan = "2", id = "mw-customcollapsible-Infobox" }) :addClass("mw-collapsible mw-collapsed") :node("<big>" .. (frame.args["SUPER Title"] or ("Super " .. ballname)) .. "</big>") ) ) -- SECTION: Identification -- Section Heading :node( p.subheading( (frame.args["SUPER Type"]), "Base Identification", "Basic Info" ) ) :node( (frame.args["SUPER Type"]) and p.s_subheading( "Super Identification" ) ) -- Type :node( p.row( (frame.args["SUPER Type"]), "Type", frame.args["Type"] ) ) :node( (frame.args["SUPER Type"]) and p.s_row( "Type", frame.args["SUPER Type"] ) ) -- Color local color = frame.args["Color"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "color" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Color") ) :node( mw.html.create( 'td' ) :node( (color ~= "") and frame:expandTemplate{ title = "Color", args = {color} } or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add color...</i>]]" ) ) ) -- Emoji local emoji = frame.args["Emoji"] or Entities.getBallAttribute( "Entities/Weapon Ball Battles/Balls/Data", ballname, "emoji" ) infobox:node( mw.html.create( 'tr' ) :node( mw.html.create( 'td' ) :css("font-weight", "bold") :node("Emoji") ) :node( mw.html.create( 'td' ) :node( (emoji ~= "") and ( emoji .. " <code>" .. p.emojiCodepointInternal(emoji) .. "</code>") or "[[Module:Entities/Weapon Ball Battles/Balls/Data|<i>Add emoji...</i>]]" ) ) ) -- SECTION: Scaling Characteristics --[[ Some data will be collected first. ]] -- (Base) Behavior local features = {} local starts = {} local scalings = {} if (featureCount > 1) then for i = 1, featureCount do table.insert(features, { frame.args["Feature "..i], " (" .. p.var(frame.args["Var "..i]) .. ")" }) table.insert(starts, { p.var( frame.args["Var0 "..i] or (frame.args["Var "..i].."₀") ) .. ": ", frame.args["Starting value "..i] }) table.insert(scalings, { p.var( "Δ"..(frame.args["Var "..i]) ) .. ": ", frame.args["Scaling "..i] }) end else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1 features[1] = {frame.args["Feature"],""} starts[1] = {"",frame.args["Starting value"]} scalings[1] = {"",frame.args["Scaling"]} end -- Super Behavior local s_features local s_starts local s_scalings -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default. local has_s_features; local has_s_starts; local has_s_scalings; if (frame.args["SUPER Feature count"]) then if s_featureCount > 1 then -- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data s_features = mw.clone(features) s_starts = mw.clone(starts) s_scalings = mw.clone(scalings) -- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors. if s_featureCount > featureCount then for i = featureCount + 1, s_featureCount do table.insert( s_features, {"",""} ) table.insert( s_starts, {"",""} ) table.insert( s_scalings, {"",""} ) end elseif s_featureCount < featureCount then for i = s_featureCount + 1, featureCount do s_features[i] = nil s_starts[i] = nil s_scalings[i] = nil end end -- Parse super data, set the has_s_... flags if triggered for i = 1,s_featureCount do if frame.args["SUPER Feature "..i] then s_features[i][1] = frame.args["SUPER Feature "..i] if (not has_s_features) then has_s_features = true end end if frame.args["SUPER Starting value "..i] then s_starts[i][2] = frame.args["SUPER Starting value "..i] if (not has_s_starts) then has_s_starts = true end end if frame.args["SUPER Scaling "..i] then s_scalings[i][2] = frame.args["SUPER Scaling "..i] if (not has_s_scalings) then has_s_scalings = true end end if frame.args["SUPER Var "..i] then s_features[i][2] = " (" .. p.var(frame.args["SUPER Var "..i]) .. ")" s_starts[i][1] = p.var( frame.args["SUPER Var0 "..i] or (frame.args["SUPER Var "..i].."₀") ) .. ": " s_scalings[i][1] = p.var( '<span style="font-style:normal">Δ</span>'..(frame.args["SUPER Var "..i]) ) .. ": " if (not has_s_features) then has_s_features = true end if (not has_s_starts) then has_s_starts = true end if (not has_s_scalings) then has_s_scalings = true end end s_features[i] = table.concat(s_features[i]) s_starts[i] = table.concat(s_starts[i]) s_scalings[i] = table.concat(s_scalings[i]) end s_features = table.concat(s_features,"<br>") s_starts = table.concat(s_starts,"<br>") s_scalings = table.concat(s_scalings,"<br>") else -- # of super features = 1? just write directly to the variable if frame.args["SUPER Var"] or frame.args["SUPER Feature"] then has_s_features = true s_features = frame.args["SUPER Feature"] end if frame.args["SUPER Var0"] or frame.args["SUPER Var"] or frame.args["SUPER Starting value"] then has_s_starts = true s_starts = frame.args["SUPER Starting value"] end if frame.args["SUPER Scaling"] then has_s_scalings = true s_scalings = frame.args["SUPER Scaling"] end end end -- finalize base data as strings now that work is otherwise complete if (featureCount > 1) then for i = 1, featureCount do features[i] = table.concat(features[i]) starts[i] = table.concat(starts[i]) scalings[i] = table.concat(scalings[i]) end features = table.concat(features,"<br>") starts = table.concat(starts,"<br>") scalings = table.concat(scalings,"<br>") else features = features[1][1] starts = starts[1][2] scalings = scalings[1][2] end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (frame.args["SUPER Feature count"]), "Base Scaling Characteristics", "Scaling Characteristics" ) ) :node( (frame.args["SUPER Feature count"]) and p.s_subheading( "Super Scaling Characteristics" ) ) -- Features :node( p.row( (has_s_features), (featureCount > 1) and "Features" or ( (frame.args["Var"]) and ("Feature (" .. p.var(frame.args["Var"]) .. ")") or "Feature" ), features ) ) :node( (has_s_features) and p.s_row( (s_featureCount > 1) and "Features" or ( (frame.args["SUPER Var"]) and ("Feature (" .. p.var(frame.args["SUPER Var"]) .. ")") or "Feature" ), s_features ) ) -- Starting values :node( p.row( (has_s_starts), (featureCount > 1) and "Starting values" or ( (frame.args["Var"]) and ("Starting value (" .. ( p.var(frame.args["Var0"] or (frame.args["Var"].."₀") ) ) .. ")") or "Starting value" ), starts ) ) :node( (has_s_starts) and p.s_row( (s_featureCount > 1) and "Starting values" or ( (frame.args["SUPER Var"]) and ("Starting value (" .. ( p.var(frame.args["SUPER Var0"] or (frame.args["SUPER Var"].."₀") ) ) .. ")") or "Starting value" ), s_starts ) ) -- Scaling :node( p.row( (has_s_scalings), "Scaling", scalings) ) :node( (has_s_scalings) and p.s_row("Scaling", s_scalings) ) -- SECTION: Career --[[ Some data will be collected first. ]] -- (Base) Games played local gamesPlayedText = {} if matchCount.total > 0 then table.insert(gamesPlayedText, matchCount.total .. " (total)") if matchCount.standard > 0 then table.insert(gamesPlayedText, matchCount.standard .. " (standard)") end end -- (Base) Winrate local winrateText = {} if matchCount.total > 0 then if matchCount.standard > 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. "% (standard)" ) end end -- Super Games played local s_gamesPlayedText = {} if s_matchCount.total > 0 then table.insert(s_gamesPlayedText, s_matchCount.total .. " (total)") if s_matchCount.standard > 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. " (standard)") end end -- Super Winrate local s_winrateText = {} if s_matchCount.total > 0 then if s_matchCount.standard > 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. "% (standard)" ) end end --[[ Begin rendering ]] -- Section Heading infobox:node( p.subheading( (isSuper), "Base Career", "Career ") ) :node( (isSuper) and p.s_subheading( "Super Career") ) -- Debuted :node( p.row( (isSuper), "Debut", frame.args["Debut"] ) ) :node( (isSuper) and p.s_row("Debut", frame.args["SUPER Debut"]) ) -- Games played :node( p.row( (isSuper), "Games played", (#gamesPlayedText > 0) and table.concat(gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Games played", (#s_gamesPlayedText > 0) and table.concat(s_gamesPlayedText, "<br>") or "<i>Not enough data...</i>" ) ) -- Winrate :node( p.row( (isSuper), "Winrate", (#winrateText > 0) and table.concat(winrateText, "<br>") or "<i>Not enough data...</i>" ) ) :node( (isSuper) and p.s_row( "Winrate", (#s_winrateText > 0) and table.concat(s_winrateText, "<br>") or "<i>Not enough data...</i>" ) ) return tostring(infobox) end function p.referenceTest() local element = mw.html.create( 'tr' ) :attr( "id","mw-customcollapsible-Infobox" ) :addClass( "mw-collapsible" ) :css( "text-align","left" ) local bigElement = mw.html.create( "div" ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface") ) :node( mw.clone(element) :attr( "colspan", "2" ) :node( "shitface2") ) return tostring(element) .. "\n" .. tostring(bigElement) .. "\n" end function p.roundedPct(num, den) return (num/den % 0.01 < 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100) end function p.title() --[[local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA" } } local cases2 = "ba" local cases3 = "ba" -- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = "true" else cases2 = "false" end -- if "ANCOVA" == frame:getParent():getTitle() then cases3 = "true" else cases3 = "false" end local detected = false detected = detected or true]] return mw.title.getCurrentTitle() end function p.ifexist() return tonumber(frame:expandTemplate{ title = "User:Helpme2222/Sandbox", args = {frame.args[1]} }) end function p.performance() for i = 1,5000000 do p.tableMake() end return end function p.performance1() local var = "" for i = 1,100000000 do if var[1] then end end return end function p.performance2() local var = "" for i = 1,100000000 do if type(var) == "table" then end end return end p.counter = 0 function p.tableBake() p.counter = p.counter + 1 return p.counter end function p.tableMake() p.counter = 0 local var = { keyword = {p.tableBake()}, kljuczslovo = {p.tableBake()}, yaoshiwenzi = {p.tableBake()}, llavepalabra = {p.tableBake()}, } return var end function p.expand() return frame:expandTemplate{ title = "User:Helpme2222/Sandbox" } end function p.title3() return mw.title.getCurrentTitle().fullText end function p.getBallAttribute() return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4]) end function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration ) if _ then return mw.text.nowiki(ballData) else return "" end end function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration) local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att] if ballData[1] then return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)] end return ballData end function p.emojiCodepointInternal( emoji ) emoji = mw.text.trim( emoji ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.emojiCodepoint() local emoji = mw.text.trim( mw.getCurrentFrame().args[1] ) local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji) ) if v then return "U+" .. string.format("%X", i ) .. " U+" .. string.format("%X", v ) else return "U+" .. string.format("%X", i ) end end function p.title2() local grammatical_cases = { ablative = 1, inessive = { ANCOVA = "ANCOVA", ["Module:Sandbox/22"] = "Module:Sandbox/22" } } local concepts = { grammatical_cases, } local PAGENAME = tostring(mw.title.getCurrentTitle()) -- local detected = false local out = {} local debugger = {} for i, group in ipairs(concepts) do if group.inessive[PAGENAME] then table.insert(debugger,"Okay, managed TRUE on "..i..":"..group.ablative) -- detected = detected or true local formattedGroup = {} for memberKey, memberRendervalue in pairs(group.inessive) do table.insert(formattedGroup, "<li>" .. memberRendervalue .. "</li>") end table.insert(out, "<h3>" .. group.ablative .. "</h3><ul>" .. table.concat( formattedGroup ) .. "</ul>" ) table.insert(out, "this sucks ass") return "this sucks ass" else table.insert(debugger,"Managed FALSE on "..i..":"..group.ablative..". Comparator: "..PAGENAME..", ") end end return debugger --[[ if detected then return tostring(detected) .. table.concat(out) end return --]] end function p.sanitizeChar(capture) -- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited if capture == "?" then return "" elseif capture == "#" then return "No. " end end function p.sanitizeTitle() local title = frame.args[1] return mw.ustring.gsub( title, "(?:\\?|\\#)", p.sanitizeChar) end function p.sanitizeTitletest2() return mw.ustring.gsub( "Ki vagy? (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)", "(?:\\?|\\#)", p.sanitizeChar()) end function p.safetyTest() if string.len(frame.args[1]) < 500 then return frame.args[1] else return "" end end --[[ function array_iter(t) local i = 0 return function () i = i + 1 return t[i] end end --]] function p.listInline() --[[ This function is a godsend! https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]] local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return mw.text.listToText( spritedTable , ", ", ", and " ) end function p.listNavbox() -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat local spritedTable = {} for i,entry in ipairs(balls[tonumber(frame.args[1])]) do spritedTable[i] = "{{Sprite|named=1|linked=1|"..entry.."}}" end return table.concat( spritedTable , " • ") end function p.listCountInternal(table) local count = 0 for index,value in ipairs( table ) do count = count + 1 end return count end function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end -- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn function p.tableTest() return balls[1][1] end function p.todaysBall() --[[ https://www.lua.org/pil/3.6.html The use of explicit indexing here is not strictly semantic; it's just to emphasize the rotation order. ]] local unixDay = math.floor(os.time()/86400) local ballTotalCount = 0 local ballCounts = {} for whichTable,subtable in ipairs(balls) do ballCounts[whichTable] = #subtable ballTotalCount = ballTotalCount + ballCounts[whichTable] end local cycleDayIndex, cycleStartingBall = unixDay % ballTotalCount, math.floor(unixDay/ballTotalCount) % ballTotalCount local cycleTodaysBall if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end --[[ debug return cycleTodaysBall .. " " .. ballTotalCount .. " " .. os.time() .. " " .. math.floor(os.time()/86400) .. " " .. cycleDayIndex .. " " .. cycleStartingBall ]] for whichTable = 1, #balls do if ballCounts[whichTable] > cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end end --]] end return p; 0e0c67tmhqteppysuoszmfi7j4szyvj 0 331037 2834968 2834835 2026-09-29T06:24:10Z Mim0502 3106283 started writing origins and conceptual development of machiavellianism 2834968 wikitext text/x-wiki {{title|Machiavellian motivation:<br>What is the motivational role of Machiavellianism?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} '''Imagine this...''' [[File:The death of Niccolò Machiavelli (1848), by Cesare Dell'Acqua.jpg|thumb|'''Figure 1'''. The death of Niccolo Machiavelli, the relevant historical figure famous for analysing quiet forms of power]] In a collaborative project, one member, Darren, quietly positions himself as the 'team organiser'. He chooses the simple, high-visibility tasks for himself and assigns the more demanding, time-heavy work to quieter members. When speaking to the supervisor, he presents the team's progress in a way that highlights his coordination and softens the significance of everyone else's contributions. By the end, he is perceived as the natural leader. You feel a mix of frustration and discouragement, not because anything openly unfairly happened, but because the imbalance was created through subtle decisions that were difficult to challenge. It leaves you with a sense of being strategically sidelined, as if your effort mattered but was never meant to be recognised. Later, while reflecting on the experience, you recall one of your colleagues briefly mention the anniversary of a historical philosopher known for analysing quiet forms of influence and power. The reference wasn't connected to your project, yet it lingers. It makes you wonder whether Darren's behaviour was intentional or simply a familiar pattern he's learned to rely on. {{ic|The target audience is a global audience, so consider making the example more applicable in that context.}} {{RoundBoxBottom}} ''author note: this overview is terribly cluttered at the moment, just jotting down notes/info I want to mention at the start then re-structuring it when I find it flows well from one point to the next'' * '''''Fig. 1:''' the death of Niccolo Machiavelli coincided with a profound moment of crisis in Italy, and immediately preceded the widespread dissemination of his work; he died having never seen his most famous work (The Prince), his realism, institutional thinking, and trust in the people further reveal a philosophy oriented toward popular participation, civic liberty and resistance to elite domination (Wills, 2021) {{ic|probably better off linking out to this detail in a Wikipedia article}}'' * ''It is important to understand dark-side traits in organisational and academic performance -'' * ''The concept of Machiavellian Motivation {{ic|use lower case for motivation}}, the core definition, machiavellianism as a personality trait - defined as using manipulation against other people's interests to achieve one's own personal goal (Aldousari & Ickes, 2021)'' * ''the central argument, what does the chapter say/prove about machiavellianism {{ic|use upper case for Machiavellianism}} : Sprangler & Tikhomirov (2025) argue that Machiavellian leaders embody "personal power"; self-focused, dominance-oriented motive that drives strategic manipulation and short-term effectiveness, and in contrast, "socialised power" leaders channel influence toward collective benefit, producing ethical sustainable organisational outcomes'' * ''drawing on the Dark Triad framework, self-determination theory, human motivation theory'' * ''motivational mechanisms, contextual application - refer back to scenario????, evidence, chapter structure'' ''Social Cognitive Theory'' ''Why do some individuals pursue influence through manipulation rather than cooperation? This question lies at the centre of Machiavellianism, a personality trait characterised by strategic manipulation and the exploitation of others for personal gain (Aldousari & Ickes, 2021). As a core component of the Dark Triad, Machiavellianism has attracted considerable attention within organisational psychology because of its implications for leadership, performance, and workplace relationships.'' ''This chapter examines the concept of Machiavellian motivation, referring to the motives and psychological processes that drive manipulative and self-serving behaviour. Although Machiavellian individuals often appear effective in competitive environments, this chapter argues that such effectiveness is largely short-lived. Specifically, it contends that Machiavellianism reflects a self-focused power orientation that may facilitate immediate gains but ultimately undermines trust, cooperation, and sustainable organisational outcomes. In doing so, the chapter challenges the notion that strategic manipulation is an effective long-term pathway to influence and success. This position aligns with Sprangler and Tikhomirov's (2025) distinction between personal power and socialised power, whereby Machiavellian leaders pursue influence for individual advantage, whereas socialised power leaders channel influence towards collective benefit, producing more ethical and sustainable organisational outcomes.'' ''Drawing on the Dark Triad framework, Social Cognitive Theory, Self-Determination Theory, and broader theories of human motivation, the chapter explores the motivational foundations of Machiavellian behaviour and the mechanisms through which it influences organisational outcomes. It first defines Machiavellianism and situates it within the wider personality literature, before examining its motivational underpinnings, reviewing empirical evidence, and discussing implications for organisations and leaders.'' Recommended length: 180 to 330 words. * [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] explains about how to edit * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] {{RoundBoxTop|theme=5}} '''Focus questions''' * How has Machiavellianism developed conceptually as a motivational construct? * What are the motivational characteristics of Machiavellianism as a trait? * What core motives drive Machiavellian individuals? * How do Machiavellian motives translate into behavioural strategies? {{RoundBoxBottom}} == Origins and conceptual development of machiavellianism == '''FQ: How has Machiavellianism developed conceptually as a motivational construct?''' ''author note: I am aware some of these sources are not peer-reviewed sources, these are just preliminary.'' * "Men are driven by necessity, and he who seeks to deceive will always find someone who will allow himself to be deceived" (Machiavelli, 1532/2017) {{ic|APA style - direct quotes need page numbers.}} * "Men are driven by two principal impulses, either by love or by fear" (Machiavelli, 1532/2017) - Machiavelli believed a ruler should ideally be both, but since human beings are unreliable, love is a weak foundation for authority. ''To answer the question, it is necessary to examine both the historical origins of Machiavellian thought and its evolution into modern psychological construct. The following section first explores Niccolò Machiavelli's assumptions about power, human nature, and political behaviour, before tracing how these ideas informed contemporary understandings of Machiavellianism as a motivational orientation characterised by strategic self-interest and the pursuit of personal influence. Particular attention is given to Machiavelli's views on human motivation, as these ideas provide the conceptual foundations for later theories linking Machiavellianism to power motives and goal-directed behaviour.'' === Historical and theoretical foundations === * Philosophies for Life. (2024): Niccolo Machiavelli - referred to as the "father of modern political philosophy", wrote the novel 'The Prince' ** separated politics from ethics ** he believed people are naturally self-interested and unreliable ** 'virtu' = skill, cleverness (courage, wisdom, decisiveness) / 'fortuna' = luck, chance ** "perception is everything" * Machiavellianism became a negative concept because readers (public & religious reactions) misinterpreted Niccolo Machiavelli's 16th century political manual, The Prince, as an endorsement of deceit rather than a realistic portrayal of how power worked (MR. BRAIN, 2025) - Machiavelli's treatment of mortality is one of the most controversial aspects of his thought, 'the ends justify the means' * He rejected the view of state existing to cultivate virtue and the good life, insisting that politics had to be considered/understood on its own terms guided by history and human psychology rather than religious doctrine or idealised notions of justice - he is often credited as an early voice of [https://www.britannica.com/topic/secularism Secularism] (Britannica) in Western political thought (Kumar, 2025) * Core pillars on his political thought = Power, manipulation and strategic pragmatism (Kumar, 2025) * "This is to be asserted in general of men, that they are ungrateful, fickle, false, cowardly, covetous" (Machiavelli, 1532/2017) * "The lion cannot defend itself against snares and the fox cannot defend itself against wolves" (Machiavelli, 1532/2017) * Connell (2022) blended intellectual history with biographical reconstruction to show how Machiavelli's end-of-life circumstances refract his lifelong preoccupation with power, fortune, and human nature ''The historical foundations of Machiavellian motivation can be traced to the political writings of Niccolò Machiavelli, widely regarded as a foundational fiture in modern political thought and best known for The Prince (1532/2017). Writing during a period of political instability, Machiavelli changed traditional views that leadership should be guided primarily by morality and virtue. Instead, he advocated a pragmatic approach to power grounded in historical observation and human psychology. Central to his thought was the belief that individuals are often self-interested and motivated by personal advantage. As Machiavelli argued, "men are driven by necessity, and he who seeks to deceive will always find someone who will allow himself to be deceived" (Machiavelli, 1532/2017, p. ). Similarly, he suggested that human behaviour is governed by powerful motivational forces, claiming that "men are driven by two principal impulses, either by love or by fear" (Machiavelli, 1532/2017, p. ). Although he maintained that rulers should ideally inspire both, he ultimately regarded fear as the more dependable basis of authority because human loyalty could nit be trusted.'' ''These assumptions formed the basis of Machiavelli's broader philosophy of power, which emphasised strategic pragmatism, adaptability, and the effective management of appearances (Kumar, 2025). His concepts of virtu and fortuna further emphasised the importance of strategic skill in navigating uncertain environments. This perspective if further captured in his famous metaphor that a ruler must learn to act as both a lion and a fox, possessing the strength to confront threats and the cunning to avoid deception. Although Machiavelli's work was originally concerned with political leadership (subsequent interpretations of The Prince contributed to the emergence of "Machiavellian" as a term assosciated with manipulation and deceit, these ideas later informed a psychological construct of Machiavellianism, which describes a tendency towards manipulation, calculated social behaviour, and the pursuit of personal goal through strategic interpersonal influence.'' === Evolution into a motivational trait === ==== Integration into modern personality models ==== * Machiavellianism as a Spectrum Trait - Persson & Kajonius (2017) provide definitive evidence that Machiavellianism is a spectrum trait; it varies continuously across individuals, with graded differences in cynicism, manipulation, and emotional detachment. It is dimensional and measurable, not categorical. * Integration into the [[wikipedia:Dark_triad|dark triad]] framework ** Paulhus & Williams (2002) establish the Dark Triad as a unified but differentiated framework of socially aversive personality; Machiavellianism, [[Narcissistic personality disorder|Narcissism personality disorder]] (Wikiversity), and [[wikipedia:Psychopathy|Psychopathy]] (Wikipedia) share a manipulative, callous interpersonal cure but diverge sharply in impulsivity, self-presentation, and strategic orientation ** Rauthmann (2012) identified stable behavioural signatures of Machiavellianism (manipulation, strategic planning, etc.) and provided structural evidence that Machiavellianism is a coherent trait, not a behavioural tendency ==== Shift toward a motivational construct ==== * Christie & Geis (1970) reached their conclusions by observing real manipulative behaviour, identifying the attitudes that predicted it, and using factor analysis to build a coherent trait scale. ** They developed the [https://openpsychometrics.org/tests/MACH-IV/ Mach-IV Scale] (MACH-IV: Machiavellianism Test), foundational self-report instrument for measuring Machiavellianism in non-clinical population, operationalising Machiavellianism as a strategic and cynical interpersonal orientation, capturing both beliefs about human nature and preferred interpersonal tactics * Establishing Machiavellianism as a Measurable Trait ** Dahling et al. (2009) demonstrated that Machiavellianism is a measurable, multidimensional trait in workplace setting through their Workplace Machiavellianism Scale; which provides strong psychometric evidence that Machiavellianism is a stable dispositional construct, not merely a situational tactic == Trait-based motivational tendencies of machiavellianism == '''FQ: What are the motivational characteristics of Machiavellianism as a trait?''' ''author note: this section, at most, is only at a fundamental level and I am still planning on breaking down into further subsections. the information/cites provided are only preliminary.'' Wilson et al. (1997) stated that Machiavellianism became a term of interest in evolutionary psychology * it has been argued that manipulative behaviour is not a single trait but rather a complex set of traits that cannot be captured by a single scale * the article argues that Machiavellianism is an adaptive, context-sensitive strategy shaped by evolutionary pressures and expressed through psychological mechanisms that favour manipulation; its effectiveness may depend on environmental context and population dynamics * Evolutionary theory explains why the trait exists Dark Triad Framework * Veselka et al. (2012) show that the Dark Triad is best understood through the HEXACO model (see Fig. 2), where low Honesty-Humility forms the common foundation and additional trait combinations differentiate machiavellianism, psychopathy, and narcissism === Cognitive architecture of machiavellian motivation === [[File:HEXACO 1.png|thumb|'''Fig. 2''' Ashton & Lee's HEXACO Model]] ==== Core Personality Characteristics (Nature of Personality) ==== * Machiavellianism strongly reflects low H-H in Ashton & Lee's (2007) HEXACO model; characterised by exploitation, entitlement, and manipulation * Christie & Geis (1970) stated machiavellians demonstrated strategic, calculating pragmatism, or the preference for instrumental, goal-focused reasoning over moral or relational considerations * High cognitive empathy and low affective empathy, or the ability to understand others' perspectives without emotional resonance (Wai & Tiliopoulos, 2012) ==== Affective and Cognitive Style (Feeling & Thoughts) ==== * Blunted moral emotions facilitate manipulation; low guilt, remorse and affective responsiveness (Ali & Chamorro-Premuzic, 2010) * Machiavellians relies on controlled processing rather than intuitive empathy; analytical social cognition (Jones & Paulhus, 2011) * Grieve & Mahar (2010) found they suppressed or modulated emotional displays to maintain influence; emotion regulation for strategi ends === Instrumental and controlled goal orientation === ==== Interpersonal Orientation (In relation to others) ==== * Dahling et al. (2008) discussed machiavellians to be skilled in navigating organisational politics and coalition building * Exploitative, transactional relational approach; wherein they only engage in relationships when instrumental returns are high (Kiazad et al., 2010) * They use charm, deception, and impression management to achieve goals revealing a manipulative interpersonal style (Christie & Geis, 1970; Jones & Paulhus, 2009) * White et al. (2024) found that machiavellianism is a meaningful predictor of intimate partner violence, particularly psychological and coercive forms * Blötner & Bergold (2023) show that cyberbullying is particularly appealing to Machiavellians because it allows them to harm other strategically while avoiding detection ==== Self-Determination Theory ==== Three basic psychological needs: Autonomy, relatedness, competence (Deci & Ryan, 2000). * McHoskey (1999) shows that Machiavellianism reflects a motivation profile dominated by extrinsic goals, low social interest, and controlled regulation ===== Extrinsic Orientation (Relatedness) ===== * Instrumental Motivation; Blötner & Bergold (2022b) demonstrated that Machiavellianism contains both approach-oriented manipulation and avoidance-oriented distrust. These facets jointly determine whether Machiavellians choose to deceive, defend, or strategically oscillate between both behaviours * External Self-concept motivation * Moss & Barbuto Jr. (2004) demonstrated that Machiavellians seek control and status using coercive, indirect and strategic tactics to influence others rather than rational or inspirational methods ===== Autonomy and Low Affiliation (and Competence) ===== * Rauthmann (2012) found that Machiavellians prefer autonomy, independence, and emotional distance ** showing low affiliative motivation and avoiding relational dependence *Wilson et al. (1996) demonstrated that Machiavellians show low need for closeness and exhibits high competence-related self-efficacy *Similarly, McHoskey (1995) reported low affiliation needs and high agentic motives such as power, control, and competence == Core motives of machiavellian individuals == '''FQ: What core motives drive Machiavellian individuals?''' David McClelland's (1965) Human Motivation Theory and Deci and Ryan's (2000) Self-Determination Theory together show that Machiavellians act the way they do because: * they are power-motivated (Human Motivation Theory) and extrinsically driven (SDT) * they pursue goals that increase control, status, and advantage, not intrinsic satisfaction * they avoid emotional closeness because affiliation and relatedness undermine strategic autonomy * they invest in competence and achievement only when it enhances influence * their motivational system is instrumental and contex-sensitive, not prosocial or self-expressive === Approach motives: power, control, and strategic gain === ==== Human Motivation Theory ==== McClelland's (1965) Human Motivation Theory: Achievement, Affiliation, Power ===== Power and Control – power motivation? - Power (nPow) ===== * Machiavellianism is positively associated with self-efficacy and job performance when individuals believe they can control outcomes, suggesting that feeling capable amplifies their motivation to pursue self-interested goals (Hafeez et al., 2024) * Closely linked to [[Motivation and emotion/Book/2013/Power motivation|Power motivation]] (Power motivation, 2013) - people who are high machs view dominance and control as core personal objectives (Christie & Geis, 1970) * Szabo et al. (2023) identify power motivation as a central driver of Machiavellian work behaviour, part of their broader pattern of controlled, extrinsic, strategic motivation, which distinguishes Machiavellianism from the other Dark Triad traits. * Aldousari & Ickes (2021) found a positive correlation between machiavellianism and an external locus of control, which reflects a worldview where powerful others and situational forces dictate outcomes. This fuels the belief system that Machiavellians' relies on manipulation and strategic behaviour as compensatory mechanisms for perceive lack of personal control ===== Reward Sensitivity - Achievement (nAch) ===== * Buravlova (2026) demonstrates that Machiavellianism is neurally characterised by string reward sensitivity and high execute control, enabling calculated manipulation and strategic social behaviours * Mercadante et al. (2026) show that managers sometimes evaluate dark-trait subordinates positively because these individuals facilitate agentic, performance-oriented goals; machiavellianism being especially rewarded due to its reward-sensitive, achievement-oriented profile * Birkás et al. (2016) demonstrate that Machiavellianism is characterised by high reward sensitivity, low punishment sensitivity, and strategic risk-taking, aligning closely with achievement motivation (nAch) and explains Machiavellians' opportunistic, reward-driven behaviour === Avoidance motives and manipulation self-efficacy === Social cognitive theory self-efficacy == Behavioural expression of machiavellian motivation == '''FQ: How do Machiavellian motives translate into behavioural strategies?''' ''author note: this section, at most, is only at a fundamental level and I am still planning on breaking down into further subsections. the information/cites provided are only preliminary.'' * Personality psychology through organisational behaviours to decision-making research underpins how machiavellian tendencies scale upward. * Jahangir et al. (2024) show that Machiavellianism is supported by a coherent psychological and neural profile" emotionally detached, strategically cognitive, clinically antagonistic, and nerually biased toward executive control over emotional processing === Strategic social influence - interpersonal behaviours === ===== Strategic Calculation ===== * Long-term Planning; This multidimensional evidence from Jahangir et al. (2024) reinforces the view of Machiavellianism as a cold, calculated social strategt rather than an impulsive or purely antisocial trait * Strategic Opacity - machiavellianism shapes decision-making not only through individual traits but through team dynamics and organisational structures as well, it is a systemic force that influences how information flows, how power is distributed and how strategic decisions are made within organisations (Matthews et al., 2022) * Calculated Manipulation; Gao et al. (2025) show that antagonistic Machiavellians achieve greater promotability not because their antagonism is inherently valued, but because certain organisational contexts reward political behaviour and misinterpret strategic self-presentation as leadership potential ===== Instrumental Sociability ===== * Social Acuity; Jones (2016) shows that machiavellians misbehave in ways that are planned, covert and opportunistic. Their antisocial behaviour is a strategic tool for achieving personal gain, not an impulsive or emotional reaction * Transactional Relationships * Utilitarian Focus; Genau et al. (2021) found that machiavellian leaders are effective only when they possess high political skill, acting as 'social lubricant' allowing machiavellian tendencies to be expressed in socially acceptable and effective ways. Without it, machiavellianism leads to dysfunctional and distrust-based leadership. === Context-shaped opportunistic behaviour - decision-making and situational behaviours === ===== Deception and Leverage ===== * Geis & Moon (1981) show that Machiavellian individuals deceive because it is instrumentally useful, emotionally easy, and cognitively manageable. Their deception is deliberate and aimed at securing advantage rather that driven by impulse or malice. * Furthermore, Blötner & Bergold's (2022a) central finding is that Machiavellian individuals are skilled "bullshit (persuasive statements made without regard for truth) producers", but not reliably more resistant to bullshit themselves * Emotional Detachment; Gao & Fang (2025) found that Machiavellianism harms adolescent wellbeing not through direct emotional distress, but through its interpersonal consequences: reduced prosocial behaviour, increased loneliness, and heightened suicidal risk * Delayed Gratification; Bratu et al. (2025) show that machiavellianism undermines academic performance because it promotes extrinsic motivation strategic lying, and low genuine engagement with learning. Machiavellian students pursue grades instrumentally rather than through mastery, which makes deception a functional (though academically harmful) strategy ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * Machiavellianism is a measurable, multidimensional spectrum trait supported by decades of psychometric work, from the Mach-IV to modern taxometric analyses confirming its continuous latent structure * Across behavioural, motivational, and neural evidence, Machiavellianism consistently aligns with high autonomy, low affiliation, and instrumental competence. ** This forms a cold, agentic, interpersonal strategy oriented toward control and reward acquisition * Empirical studies show that Machiavellians' reward sensitivity, strategic cognition, and emotional detachment hointly produce stable patterns of manipulation and opportunism, reinforcing the trait;s coherence across contexts. ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses ==References== {{Hanging indent|1= Aldousari, S. S., & Ickes, W. (2021). How is Machiavellianism related to locus of control?: A meta-analytic review. Personality and Individual Differences, 174, 110677. https://doi.org/10.1016/j.paid.2021.110677 Aldousari, S. S., & Ickes, W. (2021). How is Machiavellianism related to locus of control?: A meta-analytic review. Personality and Individual Differences, 174, 110677. https://doi.org/10.1016/j.paid.2021.110677 Ali, F., & Chamorro-Premuzic, T. 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Personality and Individual Differences, 52(7), 794–799. https://doi.org/10.1016/j.paid.2012.01.008 White, L. K., Valos, N., Xochitl, & Willis, M. L. (2024). Machiavellianism and Intimate Partner Violence Perpetration: A Systematic Review and Meta-Analysis. Trauma Violence & Abuse, 25(5). https://doi.org/10.1177/15248380241270027 Wills, M. (2021, November). ''Machiavelli, Prince of…Democracy?'' JSTOR Daily. https://daily.jstor.org/machiavelli-prince-of-democracy/ Wilson, D. S., Near, D., & Miller, R. R. (1996). Machiavellianism: A synthesis of the evolutionary and psychological literatures. Psychological Bulletin, 119(2), 285–299. https://doi.org/10.1037/0033-2909.119.2.285 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Machiavellianism]] n6pg72ct9sifzbfzwv6ni9w35pkx9hk 0 331046 2834999 2834631 2026-09-29T08:04:54Z U3253363 3106362 /* Understanding self-concept */ combining two subtitles 2834999 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check:''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. Thus, what motivates each person and how they subsequently behave depends on their self-concept. === Self-worth contingencies: How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check:''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an antecedent to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation.[[File:Multidimensional model of self-concept diagram.png|thumb|600x600px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).|center]]Repeatedly self-handicapping as motivated by threats to the self-concept may further threaten self-concept (Schwinger et al., 2022). Recurrent self-handicapping may result in patterns of underperformance which may become incorporated into one's self-concept. This is consistent with reciprocal models of self-concept development, wherein self-concept-motivated behaviours can form patterns which reciprocally shape self-concept; these will be explored later in this chapter. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' As illustrated by self-handicapping, holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions can improve wellbeing by deemphasising the extent self-worth depends on outcomes. Promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure may therefore improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === Self-determination theory: How internalising motivation shapes self-concept === [[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. [[File:Self-determination theory psychological needs.jpg|center|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation.]] Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports a developmental pathway of self-determination theory whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The reciprocal effects model: The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who one believes they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who one becomes. Ash, in our scenario, trains because not training would be inconsistent with her identity as a swimmer. Her desire to maintain congruence between behaviour and self-concept motivates her persistence even when the activity and demands are hard.{{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. 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''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] 9mw9aaco3rnakr786ez87es0gsh4ytd 2835000 2834999 2026-09-29T08:06:32Z U3253363 3106362 /* Self-worth contingencies: How self-concept can undermine motivation */ 2835000 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check:''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. Thus, what motivates each person and how they subsequently behave depends on their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check:''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an antecedent to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation.[[File:Multidimensional model of self-concept diagram.png|thumb|600x600px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).|center]]Repeatedly self-handicapping as motivated by threats to the self-concept may further threaten self-concept (Schwinger et al., 2022). Recurrent self-handicapping may result in patterns of underperformance which may become incorporated into one's self-concept. This is consistent with reciprocal models of self-concept development, wherein self-concept-motivated behaviours can form patterns which reciprocally shape self-concept; these will be explored later in this chapter. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' As illustrated by self-handicapping, holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions can improve wellbeing by deemphasising the extent self-worth depends on outcomes. Promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure may therefore improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. [[File:Self-determination theory psychological needs.jpg|center|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation.]] Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports a developmental pathway of self-determination theory whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. 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Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. 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''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] 0di2sboy2s5svab9s2h0k08wc1w7p0v 2835003 2835000 2026-09-29T08:17:57Z U3253363 3106362 /* How the self directs and energises behaviour */ Trying to reduce word count 2835003 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation.[[File:Multidimensional model of self-concept diagram.png|thumb|600x600px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).|center]]Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. [[File:Self-determination theory psychological needs.jpg|center|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation.]] Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports a developmental pathway of self-determination theory whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. 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''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] 97ghzwauqx2tts99dbcpfgxzeboxnqu 2835004 2835003 2026-09-29T08:22:16Z U3253363 3106362 /* How self-concept can undermine motivation */ Changing figure formatting 2835004 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. [[File:Self-determination theory psychological needs.jpg|center|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation.]] Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports a developmental pathway of self-determination theory whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., & Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] lpdmhjh86jhfqqrrqwzv0bog9zlf7n4 2835005 2835004 2026-09-29T08:23:33Z U3253363 3106362 /* How does motivation shape self-concept? */ 2835005 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. [[File:Self-determination theory psychological needs.jpg|center|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation.]] Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports a developmental pathway of self-determination theory whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., & Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] d3gqt7xqw2eujhimdxqmlhxf2kos039 2835006 2835005 2026-09-29T08:24:51Z U3253363 3106362 /* How internalising motivation shapes self-concept */ Fixing figure formatting 2835006 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation.]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports a developmental pathway of self-determination theory whereby fulfilling core psychological needs promotes more autonomous motivation and thus develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., & Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] teksoh7ft1lrr4owkw20ika9v7mf9q0 2835007 2835006 2026-09-29T08:28:13Z U3253363 3106362 /* How internalising motivation shapes self-concept */ 2835007 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found motivational needs (autonomy, competence, and relatedness) posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only supported as an antecedent to exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., & Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] 8e96wpv6z2sh48nuxjc129s0qzt9qz5 2835008 2835007 2026-09-29T08:33:57Z U3253363 3106362 /* The cyclical process of motivation and self-concept development */ 2835008 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the worlfd.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. [[w:Self-discrepancy_theory|Self-discrepancy theory]] (Higgins, 1987) and [[w:Regulatory_focus_theory|regulatory focus theory]] (Higgins, 1997) adopt these conceptualisations of the self as socially influenced and multi-faceted. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties, Theories of Origin and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |Self-discrepancy theory, regulatory focus theory |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, self-discrepancy theory |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach (Lietz et al., 2026). This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept (Van den Broeck et al., 2021). This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by, exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., & Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173 Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] rj7ox4zjkyzdkfrkzjy24hd2cuj3d93 2835010 2835008 2026-09-29T08:44:46Z U3253363 3106362 /* How internalising motivation shapes self-concept */ Trying to reduce word count 2835010 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves. This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?” (Wehrle & Fasbender, 2018). But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what one is motivated to do and examines how what one repeatedly does subsequently shapes self-concept. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == Understanding self-concept == Self-concept is a cognitive framework containing a system of self-knowledge. It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. Identity-based motivation theory and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - Skills needed to be successful - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by, exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] 0cwwb467zfxbdrolzhhfobtbhvtgxrl 2835011 2835010 2026-09-29T08:50:34Z U3253363 3106362 2835011 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves (Wehrle & Fasbender, 2018). This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?”. But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what people are motivated to do and examines how what people repeatedly do subsequently shapes their self-concepts. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=12}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == What is self-concept? == Self-concept is a cognitive framework containing a system of self-knowledge (Wehrle & Fasbender, 2018). It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. Identity-based motivation theory (Oyserman, 2024) and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focus positively correlated with creativity and an innovative cognitive style, whereas a prevention focus negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult-self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation, through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth, but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical multi-faceted construct and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by, exercise motivation. This result conflicts the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] rug92x3nqqec5rswsnqp2zffcz9vnnp 2835012 2835011 2026-09-29T08:56:41Z U3253363 3106362 /* How does self-concept influence motivation? */ adding oxford commas 2835012 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves (Wehrle & Fasbender, 2018). This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?”. But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world.   For Ash in the case study, being a swimmer is core to her self-concept, and succeeding in the upcoming swimming competition is an important self-concept-driven goal. This guides and motivates her behaviour of swimming early every morning.   This chapter explores how the understanding of oneself influences what people are motivated to do and examines how what people repeatedly do subsequently shapes their self-concepts. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth.{{RoundBoxTop|theme=12}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == What is self-concept? == Self-concept is a cognitive framework containing a system of self-knowledge (Wehrle & Fasbender, 2018). It is related to but unique from many of the popular “self-” constructs such as [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me” the material, social and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of the responses and evaluations of others. Identity-based motivation theory (Oyserman, 2024) and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued   |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional, ubiquitous, and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become) and ought self (who one believes others want them to be; see Table 2). Ideal and ought representations of the self set standards for one to strive toward, thus laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (see Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focuses positively correlated with creativity and an innovative cognitive style, whereas prevention focuses negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self-identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges, each modifying goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person and how they subsequently behave is shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How you feel about yourself + Standards people feel they must meet to have value as a person - The ways people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, due to how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, results in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure may not be self-attributed (Török & Szabó, 2018). A handicap generates a plausible external explanation for an outcome other than the self. This indicates that a handicapper is willing to increase the probability of failure to protect their self-concept, demonstrating that self-concept can impact motivation by hindering goal pursuit (Coudevylle et al., 2020). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical, multi-faceted construct, and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). Thus, interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concept domains, self-concept flexibility, and growth-oriented interpretations of failure to improve both wellbeing and motivation.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence, and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation. These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined one's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, by influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by exercise motivation. This result conflicts with the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable, and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change and identity-based motivation theory explains how active identities change the meaning of difficulty, whereas self-worth contingencies can undermine motivation. Together, these theories detail how self-concept influences goal direction and engagement and provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] jacu91zhvsmy6aanv9tb0hkgrc8znmu 2 331051 2835015 2834666 2026-09-29T09:38:08Z U3253363 3106362 2835015 wikitext text/x-wiki == About me == Hi, my name is [https://www.linkedin.com/in/pepper-white/ Pepper]. I am a third-year Bachelor of Science in Psychology student at the [https://www.canberra.edu.au/ University of Canberra]. I am passionate about preventative mental health care and supporting those in need. This passion motivates me in my studies and at work. === Work experience === * '''Mental Health Educator''' with [https://mieact.org.au/ Mental Illness Education ACT]. In this role, I facilitate discussions about [[Stress (psychological)|stress]], [https://www.beyondblue.org.au/mental-health/depression?gad_source=1&gad_campaignid=21934835275&gbraid=0AAAAADuibRbKbycJc05RErZ8UMCld1aGB&gclid=Cj0KCQjwkOvTBhDgARIsAKUNyRuK4CGO5WdRCpr1RDshuiDIlyi_ZwRe8ZFfvAygSGrvK0TbjFm7VtYaAkf8EALw_wcB depression], and [[wikipedia:Help-seeking|help-seeking]] with ACT school students. * '''Student Advocate''' in the [https://www.canberra.edu.au/content/myuc/home/support/student-advocacy-and-src/student-advocacy.html Student Advocacy Office] at the University of Canberra. In this role, I help my peer university students navigate university policies and procedures. === Hobbies === * [[wikipedia:Oil_painting|Oil painting]] * Figure drawing * Running * Playing water polo with the [https://www.revolutionise.com.au/dragonswp/home/ Gungahlin Dragons Water Polo Club] * Travelling == Book chapter == The book chapter I am writing is [[Motivation and emotion/Book/2026/Self-concept and motivation|Self-concept and motivation]]. The chapter navigates the question "How does self-concept relate to motivation?" I chose this topic to explore due to my fascination with how the perception of oneself can influence one's behaviour. == Social contributions == My approach for social contributions has been to edit or comment on a peer's page every time I work on my own book chapter. My hope is that this approach helps me stay engaged with others' ideas and contribute as regularly to others' work as I do my own. # I engaged with conversations in a [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261?entry_id=804001 discussion post] on Canvas about topics of interest within motivation and emotion. My goal with this post was to take inititative by being an early contributor to help build a sense of comfort and normality amongst the group. I also hope by posting some topics of interest, others may feel inspired by these focus areas, thus helping them choose their book chapter topic. # In familiarising myself with Wikiversity, I read the Spirituality and resilience 2025 book chapter. I noticed the writing required edits for readability, so I [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSpirituality_and_resilience&diff=2824208&oldid=2774009 edited the first two paragraphs] of the overview. I fixed spelling, grammar, and wording issues. This is an important fix to help capture the reader’s attention and demonstrate credibility when the audience begins reading. # I went through the topics others have chosen and the topic “Empathy and jury decision-making" caught my attention. I [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FEmpathy_and_jury_decision-making#Exploring_empathy_within_similar_contexts commented] on this book chapter with some additional ideas from my own passion about researching empathy to support this students’ topic development. # To better illustrate the multidimensional model of self-concept Marsh et al. (1992), which I refer to in my book chapter, I [https://commons.wikimedia.org/w/index.php?title=File%3AMultidimensional_model_of_self-concept_diagram.png&diff=1265404333&oldid=1265404322 created a diagram and uploaded it to Wikimedia]. # I looked through the book chapter "Empathy and jury decision-making" and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_and_jury_decision-making&diff=2826959&oldid=2824982 made edits] to the ordering of citations within a bracket in the conclusion to ensure they were alphabetical. # I read the book chapter "Falling in love" and [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FFalling_in_love#Interesting_theories_to_incorporate suggested a few theories and angles to explore] within this fascinating topic. # I read the book chapter "Charismatic leadership and follower motivation" and [[Talk:Motivation and emotion/Book/2026/Charismatic leadership and follower motivation#Interesting real-life case study to consider incorporating|suggested exploring the Jonestown cult]] as a real life case study to give insight into the extreme side of charismatic leadership and follower motivation. # To connect with others also writing 'self-' related chapters, I read the book chapter "Possible selves and goal pursuit". I identified areas which may require more research and [[Talk:Motivation and emotion/Book/2026/Possible selves and goal pursuit#c-U3253363-20260909032900-Jshottt-20260827141800|posted on their discussion forum]] some of the research I have been using which could be relevant to their chapter. # I reviewed the book chapter "Love styles and relationship satisfaction" and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FLove_styles_and_relationship_satisfaction&diff=2832497&oldid=2759229 made stylistic and grammatical edits throughout,] in particular I fixed capitalisation to sentence casing in headings and fixed the broken Wiki links. # I [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461216?entry_id=816875 suggested an easy and new way to check your chapter's word count] in a Canvas discussion board. # In reviewing a chapter of interest to me, Volunteer counsellor motivation: Difference between revisions, I noticed the scenario being used did not involve a made up person or factual story, it instead had an introduction to a theory. To help the author catch the reader's attention, I [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FVolunteer_counsellor_motivation&diff=2833605&oldid=2832761 added a suggestion for a scenario] which involves a concrete case and a real person to help the reader connect immediately with the chapter. # I read the drafted book chapter called Body neutrality and emotional wellbeing and noticed headings need to be [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FBody_neutrality_and_emotional_well-being&diff=2833988&oldid=2830006 adjusted to sentence case,] so I assisted the author by fixing this up. # To better illustrate goal orientation theory when I reference it in my book chapter, I tried [[c:File:Mastery_versus_performance_orientation_illustration.jpg|making a diagram using Gemini and uploaded it to WikiMedia]]. # I tried searching for a clipart-style image to support readers' understanding of self-determination theory's psychological needs but could not find one. So, I used [[c:File:Self-determination_theory_psychological_needs.jpg|Gemini to help me create a diagram and uploaded it to WikiMedia]]. # I read the textbook chapter Relatedness motivation in self-determination theory and noticed the scenario warranted a few minor edits for readability, so [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FRelatedness_motivation_in_self-determination_theory&diff=2834093&oldid=2833064 I made some wording adjustments and added a more complete caption] to the scenario figure. # I [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/463438 invited feedback on my book chapter via the Canvas discussion] forum and committed to returning feedback to all who provide feedback on my book chapter. # I [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FGuided_meditation_and_emotion_regulation&diff=2834575&oldid=2757490 fixed in-text citation errors] in the textbook chapter Guided meditation and emotion regulation to ensure they were APA 7 style. # After inviting feedback on my book chapter, I in turn gave feedback to someone who reviewed my chapter, via their chapter's [[Talk:Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment#c-U3253363-20260927103400-Feedback|discussion forum]]. # I also [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FImmersive_therapy_for_PTSD_treatment&diff=2834661&oldid=2834467 made direct edits] in Immersive therapy for PTSD treatment to improve readability and I added inter-Wiki links. # I engaged with someone who provided feedback on my chapter via [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/463438?entry_id=821481 Canvas discussions] to thank them and notify them I had left some edits in exchange. # After inviting feedback on my book chapter, I in turn [[Talk:Motivation and emotion/Book/2026/Empathy and jury decision-making#c-U3253363-20260927110400-Feedback|gave feedback]] to someone who reviewed my chapter, via their chapter's discussion forum. # I [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FImpulsivity_versus_sensation-seeking&diff=2835013&oldid=2834774 edited a peer's book chapter] by adding a table as an option to communicate dimensions of a model instead of using a long-winded and hard-to-organise sentence that they had. # I updated the peer referred to in the above social contribution about this change via their [[Talk:Motivation and emotion/Book/2026/Impulsivity versus sensation-seeking#c-U3253363-20260929093300-Suggestion to add a table|discussion forum]] to provide some context and a further suggestion to perhaps then shorten the tricky long sentence as the table now does the heavy lifting. hw89ic9el01u885k4tpx5qpgzze5ztr 0 331100 2834975 2834659 2026-09-29T07:25:34Z U3239236 3106753 2834975 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, research suggests that higher emotional intelligence is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how researchers define and measure emotional intelligence and wellbeing. Emotional awareness, understanding, regulation, stress coping, and interpersonal functioning may help explain why emotional intelligence relates to wellbeing. However, the different components of emotional intelligence may not contribute equally, and findings can differ between trait and ability models of EI. Research into affective intelligence interventions also suggests that some emotional abilities may be developed, creating possible opportunities to support wellbeing, although improvements in EI should not be assumed to automatically produce improvements in wellbeing. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also Internal Links == * [[Emotional intelligence]] * Emotional regulation * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * Psychological wellbeing * Coping * Stress * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] jh50nwoeilul2rvpx17tk2rrnggpuf6 2834976 2834975 2026-09-29T07:29:36Z U3239236 3106753 reword conclusion with grammarly to shorten words 2834976 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] tujo64swxynavbecp50ue03vs41vqis 2834978 2834976 2026-09-29T07:31:44Z U3239236 3106753 2834978 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? <quiz display=simple> {Type the question here... |type="()"} + The correct answer. - Wrong or misleading answer. - Wrong or misleading answer. - Wrong or misleading answer. </quiz> {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] 5f11w03ymsatqz8c7ffxjkcqeseqop7 2834980 2834978 2026-09-29T07:33:18Z U3239236 3106753 2834980 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display=simple> {What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] 65oqi1kw8kbege3rtz2wmp3pkiltc1x 2834983 2834980 2026-09-29T07:35:27Z U3239236 3106753 used chatgpt to generate a quiz code to use 2834983 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display=simple> {Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display=simple> {Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] gu59jcut3ubu9aa9pfy1322qeon9fq9 2834984 2834983 2026-09-29T07:36:24Z U3239236 3106753 added two more quizzes 3 and 4 2834984 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] 2nxnvjci7elo39la69o3lrziknxkce7 2834986 2834984 2026-09-29T07:38:03Z U3239236 3106753 2834986 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. Each branch may contribute to wellbeing in a different way. When people accurately notice emotions, they become more aware of what they and others are feeling. Using emotions to guide thinking can shape attention and judgement. Understanding emotions helps people make sense of why feelings arise and how they may change over time. Managing emotions involves regulating emotional responses, which may be particularly useful during difficult situations, such as when Sophie takes a moment to calm herself before talking to her friend.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] dqj1ppqp1d4iyc3d69xory45uiau2z3 2834989 2834986 2026-09-29T07:40:46Z U3239236 3106753 2834989 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to the ways people influence their emotional experiences and responses. Regulation does not necessarily mean removing or suppressing negative emotions. Instead, it can involve changing how a situation is interpreted, deciding how an emotion should be expressed, or choosing how to respond to it. Different regulation strategies may have different consequences for wellbeing. Cognitive reappraisal, for example, involves changing the way a situation is interpreted in order to change its emotional impact. Suppression, in contrast, involves reducing the outward expression of an emotion without necessarily changing the underlying emotional experience. Emotional intelligence may help people identify which regulation strategy is appropriate for a particular situation rather than responding automatically. This may provide one explanation for the relationship between emotional intelligence and wellbeing ( MacCann et al., 2026). People who are better able to recognise and understand their emotions may also be better positioned to regulate difficult emotional experiences. However, evidence is needed to establish whether emotion regulation actually explains, or mediates, the relationship between emotional intelligence and wellbeing rather than simply being associated with both. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] du9tt34x18n73gwfvnj14eirrunalzx 2834994 2834989 2026-09-29T07:56:16Z U3239236 3106753 shortened emotion regulation paragraph 2834994 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student who has several assessments due while also dealing with conflict with a close friend. After receiving a disappointing mark, Sophie initially feels frustrated, embarrassed, and overwhelmed. Rather than immediately reacting, she recognises that disappointment and stress are influencing how she is thinking about the situation. She takes some time to regulate her emotions, considers why the result affected her so strongly, and later talks calmly with her friend about the conflict. Another student experiencing the same circumstances might struggle to identify what they are feeling, become increasingly overwhelmed, or react impulsively. Why might people respond so differently to similar emotional situations? One possible explanation involves emotional intelligence the capacity to perceive, understand, use, and regulate emotion. Emotional intelligence may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than simply experiencing positive emotions or avoiding negative ones. It concerns how people experience and manage emotions and their broader capacity to function psychologically. Understanding the relationship between emotional intelligence and emotional wellbeing is important because difficult emotions are unavoidable. Psychological research can help explain whether emotionally intelligent abilities support wellbeing, which components of emotional intelligence may be particularly important, and the psychological processes that could explain this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to how people influence their emotional experiences and responses. Rather than eliminating negative emotions, regulation can involve changing how a situation is interpreted or choosing how to respond. Different strategies may have different consequences for wellbeing. Cognitive reappraisal changes the interpretation of a situation to alter its emotional impact, however suppression reduces emotional expression without necessarily changing the underlying experience. Emotional intelligence may help people select appropriate regulation strategies, potentially explaining its relationship with wellbeing (MacCann et al., 2026). However, further evidence is needed to determine whether emotion regulation actually mediates the relationship between emotional intelligence and wellbeing. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] 6vpjn7n2xdcoosqz4lpoyzhjgm49zyt 2834995 2834994 2026-09-29T07:58:41Z U3239236 3106753 shortened scenario to cut down 100 words 2834995 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student managing several assessments and conflict with a close friend. After receiving a disappointing mark, she feels frustrated, embarrassed, and overwhelmed. Rather than reacting immediately, she recognises her emotions, takes time to regulate them, and later talks calmly with her friend. Another student in the same situation might become overwhelmed or react impulsively. Why might people respond differently to similar emotional situations? One explanation involves emotional intelligence (EI), the capacity to perceive, understand, use, and regulate emotions. EI may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than experiencing positive emotions or avoiding negative ones; it also concerns how people experience and manage emotions. Understanding the relationship between EI and wellbeing can help explain which emotional abilities may support wellbeing, which components are particularly important, and the psychological processes underlying this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research supports a positive association between emotional intelligence and wellbeing. Higher EI is generally associated with greater subjective wellbeing and lower psychological distress, although the strength of these relationships varies across studies. Associations also differ according to how EI is conceptualised and measured, with stronger relationships often reported for self-report measures than performance-based ability measures (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to how people influence their emotional experiences and responses. Rather than eliminating negative emotions, regulation can involve changing how a situation is interpreted or choosing how to respond. Different strategies may have different consequences for wellbeing. Cognitive reappraisal changes the interpretation of a situation to alter its emotional impact, however suppression reduces emotional expression without necessarily changing the underlying experience. Emotional intelligence may help people select appropriate regulation strategies, potentially explaining its relationship with wellbeing (MacCann et al., 2026). However, further evidence is needed to determine whether emotion regulation actually mediates the relationship between emotional intelligence and wellbeing. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] ixzbfeu2m903gkam2od30t0hbqip1a2 2834997 2834995 2026-09-29T08:00:37Z U3239236 3106753 cut down more words from what does the evidence tell us? with grammarly 2834997 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student managing several assessments and conflict with a close friend. After receiving a disappointing mark, she feels frustrated, embarrassed, and overwhelmed. Rather than reacting immediately, she recognises her emotions, takes time to regulate them, and later talks calmly with her friend. Another student in the same situation might become overwhelmed or react impulsively. Why might people respond differently to similar emotional situations? One explanation involves emotional intelligence (EI), the capacity to perceive, understand, use, and regulate emotions. EI may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than experiencing positive emotions or avoiding negative ones; it also concerns how people experience and manage emotions. Understanding the relationship between EI and wellbeing can help explain which emotional abilities may support wellbeing, which components are particularly important, and the psychological processes underlying this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research suggests that higher EI is linked to greater subjective wellbeing and lower psychological distress. However, the strength of this relationship varies depending on how EI is defined and measured (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to how people influence their emotional experiences and responses. Rather than eliminating negative emotions, regulation can involve changing how a situation is interpreted or choosing how to respond. Different strategies may have different consequences for wellbeing. Cognitive reappraisal changes the interpretation of a situation to alter its emotional impact, however suppression reduces emotional expression without necessarily changing the underlying experience. Emotional intelligence may help people select appropriate regulation strategies, potentially explaining its relationship with wellbeing (MacCann et al., 2026). However, further evidence is needed to determine whether emotion regulation actually mediates the relationship between emotional intelligence and wellbeing. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Components of emotional intelligence and their potential relevance to emotional wellbeing'' {| class="wikitable" |+ !EI component !What it involves !Potential relevance to wellbeing |- |Emotion perception |Recognising emotions in oneself and others |May support emotional awareness and identification of emotional needs |- |Using emotions |Using emotional information to support thinking |May influence attention, judgement, and problem-solving |- |Emotion understanding |Understanding the causes, changes, and combinations of emotions |May help people make sense of emotional experiences and select appropriate responses |- |Emotion management |Regulating emotional responses in oneself and responding to others |May support coping with difficult emotions and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] dhs7bya7xu31dd2sh4yhvys6r2s24ao 2834998 2834997 2026-09-29T08:04:03Z U3239236 3106753 cut down to 3996 words 2834998 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student managing several assessments and conflict with a close friend. After receiving a disappointing mark, she feels frustrated, embarrassed, and overwhelmed. Rather than reacting immediately, she recognises her emotions, takes time to regulate them, and later talks calmly with her friend. Another student in the same situation might become overwhelmed or react impulsively. Why might people respond differently to similar emotional situations? One explanation involves emotional intelligence (EI), the capacity to perceive, understand, use, and regulate emotions. EI may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than experiencing positive emotions or avoiding negative ones; it also concerns how people experience and manage emotions. Understanding the relationship between EI and wellbeing can help explain which emotional abilities may support wellbeing, which components are particularly important, and the psychological processes underlying this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research suggests that higher EI is linked to greater subjective wellbeing and lower psychological distress. However, the strength of this relationship varies depending on how EI is defined and measured (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to how people influence their emotional experiences and responses. Rather than eliminating negative emotions, regulation can involve changing how a situation is interpreted or choosing how to respond. Different strategies may have different consequences for wellbeing. Cognitive reappraisal changes the interpretation of a situation to alter its emotional impact, however suppression reduces emotional expression without necessarily changing the underlying experience. Emotional intelligence may help people select appropriate regulation strategies, potentially explaining its relationship with wellbeing (MacCann et al., 2026). However, further evidence is needed to determine whether emotion regulation actually mediates the relationship between emotional intelligence and wellbeing. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Emotional intelligence components and wellbeing'' {| class="wikitable" !EI component !What it involves !Relevance to wellbeing |- |'''Emotion perception''' |Recognising emotions |Supports emotional awareness |- |'''Using emotions''' |Using emotions to guide thinking |Supports attention and judgement |- |'''Emotion understanding''' |Understanding causes and changes in emotions |Supports understanding and responding |- |'''Emotion management''' |Regulating emotional responses |Supports coping and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] 506cbfwma39kjzmd3cjnkp1xvf5ys72 2835001 2834998 2026-09-29T08:09:39Z U3239236 3106753 2835001 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student managing several assessments and conflict with a close friend. After receiving a disappointing mark, she feels frustrated, embarrassed, and overwhelmed. Rather than reacting immediately, she recognises her emotions, takes time to regulate them, and later talks calmly with her friend. Another student in the same situation might become overwhelmed or react impulsively. Why might people respond differently to similar emotional situations? One explanation involves emotional intelligence, the ability to perceive, understand, use, and regulate emotions. EI may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than experiencing positive emotions or avoiding negative ones; it also concerns how people experience and manage emotions. Understanding the relationship between EI and wellbeing can help explain which emotional abilities may support wellbeing, which components are particularly important, and the psychological processes underlying this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. ==== Emotional intelligence and negative wellbeing ==== EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. ===== Trait and ability emotional intelligence ===== Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing ====== '''What does the evidence tell us?''' ====== Overall, research suggests that higher EI is linked to greater subjective wellbeing and lower psychological distress. However, the strength of this relationship varies depending on how EI is defined and measured (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. ==== Emotional awareness and perception ==== Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} ==== Understanding emotions ==== Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. ===== Emotion regulation ===== Emotion regulation refers to how people influence their emotional experiences and responses. Rather than eliminating negative emotions, regulation can involve changing how a situation is interpreted or choosing how to respond. Different strategies may have different consequences for wellbeing. Cognitive reappraisal changes the interpretation of a situation to alter its emotional impact, however suppression reduces emotional expression without necessarily changing the underlying experience. Emotional intelligence may help people select appropriate regulation strategies, potentially explaining its relationship with wellbeing (MacCann et al., 2026). However, further evidence is needed to determine whether emotion regulation actually mediates the relationship between emotional intelligence and wellbeing. ====== Coping with stress ====== Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. ====== Social relationships and support ====== Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024).. Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). ==== '''Using emotions''' ==== Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. ==== Emotion understanding ==== Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). ===== Emotion regulation and management ===== Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). ====== Comparing the components ====== Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 2.''' ''Emotional intelligence components and wellbeing'' {| class="wikitable" !EI component !What it involves !Relevance to wellbeing |- |'''Emotion perception''' |Recognising emotions |Supports emotional awareness |- |'''Using emotions''' |Using emotions to guide thinking |Supports attention and judgement |- |'''Emotion understanding''' |Understanding causes and changes in emotions |Supports understanding and responding |- |'''Emotion management''' |Regulating emotional responses |Supports coping and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). ==== Emotional intelligence interventions ==== EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. ===== Do improvements in emotional intelligence improve wellbeing? ===== An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). ====== Limitations and practical implications ====== Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key... Technium Education and Humanities . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. Journal of Happiness Studies, 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. Healthcare (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. Emotion Review, 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. Journal of Happiness Studies, 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. Journal of Intelligence, 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. Personality and Individual Differences, 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. The Journal of Positive Psychology, 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. Frontiers in Human Neuroscience, 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] d666hyjucqzd5tqg1bpr4qbulhbqm1y 0 331103 2834946 2834807 2026-09-29T01:16:50Z U3247927 3005952 /* References */ 2834946 wikitext text/x-wiki {{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Couple cuddling on the couch with a movie.png|right|250x250px]] '''Scenario''' After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}} a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that "Romantic movies make relationships looks so effortless. I wish real life worked like that" {{RoundBoxBottom}} Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ? As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function. Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner & Wilson, 2013). Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021). Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships. {{RoundBoxTop|theme=3}} '''Focus questions''' * What are romantic beliefs and relationship expectations and why are they important? * What romantic ideals and relationship messages are commonly portrayed in romantic entertainment? * How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships? * What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge? {{RoundBoxBottom}} == Romantic beliefs and relationship expectations == ==== What is love? ==== [[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia & Dunne, 2024)'''.''''' ==== Romantic relationships ==== Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships. ==== Romantic ideals / romanticism ==== Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher & Metts, 1989; Knox & Sporakowski, 1968, as cited in Sprecher & Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle ("love finds a way"), the existence of a soulmate ("one and only"), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner & Wilson, 2013''). * '''Love finds a way - belief that love can overcome all obstacles''' * '''One and only - belief that we have "soulmates"''' * '''Idealisation - belief that a "true love" will nearly be perfect''' * '''Love at first sight - belief when you meet the right person, you will know''' Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher & Metts, 1989). == Romantic ideals portrayed in entertainment == Romantic ideals are influenced by a range of social and cultural factors; however, these beliefs do not develop in isolation. Entertainment media play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Studies examining romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (2013) found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of romantic partners. Furthermore, romantic comedies frequently depict themes associated with romantic ideals, including love at first sight, soulmates, idealised partners, and the belief that love can overcome obstacles. These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships. Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics. Romantic films / rom coms * soulmates * grand gestures * destiny * love conquers all * attractive partners * conflict resolved quickly * overcoming obstacles * happily-ever-after ending === Reality dating television === * accelerated relationship development * attraction and compatibility * dating as entertainment === Popular music === * love songs reinforce cultural ideas about passion, intimacy and commitment == Psychological mechanisms of media influence == '''Social exchange theory (SET)''' In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives. Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships. === Romantic scripts and schemas === === Cultivation theory === === Social cognitive theory === Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.<quiz display="bold"> {A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?: |type="()"} - She has developed a parasocial relationship with the fictional husband + She is comparing her relationship to the idealised rewards and expectations portrayed in the media - She is experiencing attachment insecurity caused by watching romantic television </quiz> == Benefits and drawbacks of romantic entertainment == Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014). === Potential benefits === === Potential drawbacks === ==== Social Exchange Theory and relationship expectations ==== Use this heading structure: * [[#Overview|Overview]] * * What is love * * * 3 to 6 major headings tailored to the topic; can have sub-headings, but: ** avoid having only one sub-heading ** provide an introductory paragraph before breaking into sub-sections * [[#Conclusion|Conclusion]] * See also * References * External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner & Kretz, 2021). Following the same beliefs as romantic comedy films. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== [[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words === To what extent should people use romantic entertainment as a guide for real-life relationships? === This is essentially your concluding discussion{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33. Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456. Chen, Y., Xia, M., & Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442 Hefner, V., & Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290 Hefner, V., & Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697 Lippman, J. R., Ward, L. M., & Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034 Knee, C. R. (1998). Implicit Theories of Relationships: Assessment and Prediction of Romantic Relationship Initiation, Coping, and Longevity. Journal of Personality and Social Psychology, 74(2), 360–370. https://doi.org/10.1037/0022-3514.74.2.360 Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318 Sprecher, S., & Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009 Van Monsjou, E., & Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Love]] [[Category:Motivation and emotion/Book/Media]] [[Category:Motivation and emotion/Book/Relationships]] 4fyls8wq2ou6hh5j9ej9ds9kmiv2wu4 0 331201 2834941 2834203 2026-09-29T01:00:18Z StudentUC2026 3105303 Heading edits 2834941 wikitext text/x-wiki {{title|Expectancy-value theory of educational motivation<br>''What is expectancy-value theory and how can it be applied to understand and enhance educational motivation?''}} __TOC__ ==Overview == {| style="width:100%; background:#FFE1F4; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#F9C3E5; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Imagine this scenario |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | <div style="background-color:#F6F4E7;{{Text color default}}; padding:8px; width:150px; float:right; text-align:center;"> [[File:Motivation Laptop.svg|150px|alt=A student reflecting on their motivation to learn]] <br> <span style="line-height:1.2; display:block; margin-top:2px;"> '''Figure 1'''. A student reflecting on their motivation to learn. </span> </div> You're sitting at your desk, working through a short maths quiz before dinner. The first question looks familiar, yet you pause, unsure whether you can solve it. You read it again, feeling a small wave of hesitation and mild frustration as you try to work out where to begin (see figure 1). After a moment, you recall solving a similar problem last week, and your teacher mentioned you were on the right track. This memory gives you a small boost of confidence, enough to try the first step. As you work through the question, you start thinking about why you’re even doing this quiz. You remind yourself that completing these tasks helps you prepare for the upcoming test and supports your longer‑term academic goals. This shift in mindset increases your confidence and clarifies the importance of the task. The quiz feels more manageable, and you start working through the first question. With this renewed sense of direction, you continue working through the quiz. This scenario demonstrates how expectancy and task value shape a learner’s motivation during routine academic activities. Expectancy-value theory provides a clear framework for understanding how changes in confidence and task value affect a learner’s motivation. This theory proposes that motivation increases when learners feel capable of succeeding and when they view a task as important, useful, or aligned with their goals. |} The scenario above shows how small shifts in confidence and perceived value can make everyday learning tasks feel either overwhelming or manageable. Many students experience fluctuating motivation during routine academic tasks, which can make learning feel overwhelming or discouraging. Expectancy-value theory helps explain this pattern by showing how confidence (expectancy beliefs) and perceived importance or usefulness (task values) shape a learner’s motivation. This theory provides a [[wikipedia:Social_cognition|social-cognitive]] explanation of why students choose to engage in learning activities and offers evidence-based strategies to strengthen both expectancy beliefs and task values. The following sections outline these core components and show how to apply them in educational settings to support more adaptive learning behaviours. {| style="width:100%; background:#CBF2D3; color:black; border-radius:12px; border:0;" |- ! style="background:#96E25B; color:black; border-radius:12px 12px 0 0; border:0;" | Focus questions |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * What is expectancy-value theory? * How do social‑cognitive models explain educational motivation? * How do expectancy beliefs influence educational motivation? * How do task values influence educational motivation? * How can expectancy-value theory be used to enhance motivation in educational settings? |} == Expectancy-value theory == [[File:Situated Expectancy-Value Theory SEVT Diagram.png|thumb|right|450px|Figure 2. Conceptual diagram illustrating expectancy‑value theory (EVT). Diagram created by author.]] * Introduce Eccles & Wigfield (2002) and explain that expectancy-value theory (EVT) was developed within achievement motivation research, focusing on how students’ expectations for success and the value they place on tasks influence their motivation and choices (see Figure 2). * Eccles and Wigfield (2000, 2002) describe expectancy-value theory as a major explanation of achievement motivation. The theory proposes that people are motivated when they believe they can succeed at a task and when they see the task as worthwhile. For example, a student is more likely to study for a maths test if they feel confident in their ability and if the test seems important, enjoyable, or useful. * Outline the key components of expectancy-value theory by explaining that motivation is shaped by two main factors: a person’s expectations for success and the value they attach to the task. Expectations reflect beliefs about one’s ability to perform well, while task value captures how important, enjoyable, or useful the activity feels. Together, these factors influence the choices students make and the effort they invest. * Although EVT is widely used in educational motivation research, scholars have noted several limitations. Pintrich (2003) argues that the theory places too much emphasis on rational decision-making and does not fully account for the emotional and social factors that influence student motivation. Other researchers highlight that EVT provides limited explanation of how expectancies and values develop over time or why they differ between students (Flake et al., 2015). These critiques suggest that EVT may not fully capture the complexity of real-world motivation. As shown in Table 1, expectancy-value theory consists of five core components that influence students’ motivation. '''Table 1.''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! style="background:#B3BBE4;{{Text color default}};" | Component ! style="background:#B3BBE4;{{Text color default}};" | Definition ! style="background:#B3BBE4;{{Text color default}};" | Example in education |- | style="background:#CEEEF4;{{Text color default}};" | '''Expectancy''' | style="background:#CEEEF4;{{Text color default}};" | Belief about ability to succeed | style="background:#CEEEF4;{{Text color default}};" | "I can pass this exam if I study." |- | style="background:#CEEEF4;;{{Text color default}};" | '''Intrinsic value''' | style="background:#CEEEF4;{{Text color default}};" | Enjoyment or interest in the task | style="background:#CEEEF4;{{Text color default}};" | "I enjoy learning psychology." |- | style="background:#CEEEF4;{{Text color default}};" | '''Utility value''' | style="background:#CEEEF4;{{Text color default}};" | Usefulness for future goals | style="background:#CEEEF4;{{Text color default}};" | "This unit is required for my degree." |- | style="background:#CEEEF4;{{Text color default}};" | '''Attainment value''' | style="background:#CEEEF4;{{Text color default}};" | Importance to identity | style="background:#CEEEF4;{{Text color default}};" | "Being good at maths matters to me." |- | style="background:#CEEEF4;{{Text color default}};" | '''Cost''' | style="background:#CEEEF4;{{Text color default}};" | Negative aspects of engagement | style="background:#CEEEF4;{{Text color default}};" | "This assignment is stressful and time-consuming." |} ==How do social-cognitive models explain educational motivation?== {{expand}} === Expectancy beliefs === * Expectancy beliefs refer to students’ judgments about how capable they are of successfully performing a task. In social cognitive models, these beliefs are central because they strongly predict academic choices, effort, and persistence. Eccles and Wigfield (2002) emphasise that expectancies for success are shaped by students’ prior achievement, self-concept, and perceptions of task difficulty. When students believe they can succeed, they are more likely to engage with learning activities and persist through challenges. * Introduce Bandura (1997), Pajares (1996), and Schunk and Pajares (2009) as key social cognitive researchers who demonstrate that expectancy beliefs strongly predict students’ effort, persistence, and academic performance. * Expectancy beliefs help explain educational motivation because students who believe they can succeed are more likely to engage with learning tasks, persist through challenges, and choose activities that support academic growth. This shows how social cognitive models account for differences in motivation across students with similar abilities. * Expectancy beliefs have been critiqued for focusing heavily on cognitive judgements and underestimating emotional and contextual influences on motivation. Research shows that confidence alone does not fully explain student engagement, particularly when anxiety, stress, or classroom climate reduce their willingness to participate (Pajares, 1996; Bandura, 1997). === Task value === * Introduce Eccles and Wigfield (2000, 2002) by explaining that task values describe how important, enjoyable, or useful students perceive a learning activity to be. * Eccles and Wigfield (2000, 2002) explain that task values include intrinsic value (enjoyment), utility value (usefulness), attainment value (importance), and cost, which refers to the perceived negative aspects of engaging in a task. These values shape how worthwhile students believe a learning activity is. * Research shows that students are more motivated when they find a task enjoyable or personally meaningful (Eccles & Wigfield, 2000). Utility value also increases motivation, with studies demonstrating that students work harder when they believe a task will support their future goals (Hulleman & Harackiewicz, 2009). * More recent work highlights the importance of cost, showing that high perceived effort, anxiety, or time demands can reduce motivation even when other values are high (Flake et al., 2015). This demonstrates that negative perceptions can undermine engagement despite strong intrinsic or utility value. * Task values help explain educational motivation because students are more likely to engage with learning activities, they find enjoyable, important, or useful. This shows how social cognitive models account for differences in motivation by focusing on how students evaluate the worth of academic tasks. * Task value research has been critiqued for focusing mainly on cognitive evaluations and overlooking how emotions, social pressures, and classroom climate shape students’ perceptions of value. Researchers also note that the different types of value can blur together in real learning situations, which makes it challenging to distinguish them clearly (Flake et al., 2015; Eccles, 2005{{ic|Use alphabetical order}}). Table #. Table caption goes here {| style="width:100%; background:#DBEAFA; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#B2E4FF; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Whats happening in this space? |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | '''The renaissance of cost'''<br> While early EVT research focused heavily on confidence and utility, contemporary studies increasingly emphasise the cost component. Recent work shows that psychological strain, performance anxiety, and time investment, captured across multidimensional cost categories such as effort, emotional strain, and opportunity costs, can meaningfully undermine high task values in demanding subjects like mathematics (Flake et al., 2015; Perez et al., 2014; Gaspard et al., 2015).<br> '''Real time and dynamic motivation'''<br> Motivation is now often viewed as a shifting, context-sensitive process rather than a static mindset. Recent studies show that students’ values and expectancies vary across lessons and classroom environments, influenced by factors such as perceived support, peer interactions, and task characteristics, which aligns with EVT’s emphasis on how beliefs and values develop within specific learning contexts (Eccles & Wigfield, 2020; Lazarides & Raufelder, 2017; Dietrich et al., 2017).<br> '''Utility value interventions'''<br> Rather than a static mindset, academic motivation is increasingly understood as a dynamic, context-sensitive process. Grounded in contemporary situated expectancy-value theory (SEVT), empirical evidence demonstrates that students’ expectancies for success and task values show meaningful micro-fluctuations across days, lessons, and distinct learning spaces (Eccles & Wigfield, 2020; Dietrich et al., 2017). These real-time shifts are shaped by immediate environmental inputs, including classroom climate, peer dynamics, and specific task designs (Lazarides & Raufelder, 2017).<br> '''The move towards situated expectancy-value theory'''<br> Early EVT (Eccles et al., 1983) focused on stable beliefs about competence and task value. Later refinements clarified how social influences (teachers, parents, and peers) shape these beliefs. SEVT (2020) reframes motivation as dynamic and situated, showing that expectancies and values shift in response to classroom context, feedback, identity, and social interactions. |} == How can expectancy-value theory be used to enhance motivation in educational settings? == * Strengthening expectancy beliefs: EVT can enhance motivation by increasing students’ expectancy beliefs through clear feedback, modelling effective strategies, and providing structured support that builds confidence in their ability to succeed (Bandura, 1997; Schunk & Pajares, 2009). * Enhancing task values: Teachers can increase task values by emphasising the relevance, usefulness, and personal meaning of learning activities, helping students connect academic tasks to future goals and real‑world applications (Eccles & Wigfield, 2002; Hulleman & Harackiewicz, 2009). * Reducing perceived cost: EVT can be applied by reducing perceived cost, such as anxiety, workload pressure, or time demands, through supportive classroom climates, manageable task structures, and opportunities for autonomy (Flake et al., 2015). * A practical example of EVT in educational settings is the utility‑value intervention, where students reflect on how a learning task connects to their personal goals or future plans. This approach has been shown to boost motivation and academic performance, particularly for students with lower initial expectancy beliefs (Hulleman & Harackiewicz, 2009). * A critique of applying EVT in classrooms is that increasing task values or expectancy beliefs may not be effective if students face structural barriers such as poor teaching quality, limited resources, or high stress. EVT interventions can also be difficult to sustain over time without consistent support (Eccles & Wigfield, 2002). == Knowledge checkpoint == {| style="width:50%; background:#F4CECE; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#E07F87; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | <big>Quiz</big> |- | style="border-radius:0 0 12px 12px; padding:2px; border:0;" | <div style="margin-top:-15px;"> <quiz display=simple> {Which component of expectancy-value theory refers to a student's belief about their ability to succeed? |type="()"} + Expectancy - Intrinsic value - Utility value - Cost {Which value component {{ic|of what?}} reflects how enjoyable or interesting a task is? |type="()"} - Attainment value + Intrinsic value - Utility value - Cost </quiz> </div> |} ==Conclusion== • Expectancy beliefs play a central role in educational motivation by shaping students’ confidence in their ability to succeed, influencing effort, persistence, and engagement (Eccles & Wigfield, 2002). • Task values determine how important, interesting, or useful a learning activity feels, guiding students’ willingness to invest time and energy in academic tasks (Wigfield & Eccles, 2000). • Expectancy-value theory shows that motivation increases when students believe they can succeed and when they view tasks as meaningful, relevant, or aligned with personal goals (Eccles & Wigfield, 2020). • Applying EVT in educational settings involves strengthening both expectancy beliefs and task values through supportive feedback, meaningful learning activities, and reduced perceived cost (Hulleman & Harackiewicz, 2009; Flake et al., 2015). • Together, these components provide a practical framework for understanding and improving student motivation in everyday classroom contexts (Eccles & Wigfield, 2002). {| style="width:100%; background:#FEFB9F; color:black; border-radius:12px; border:0;" |- ! style="background:#FFD208; color:black; border-radius:12px 12px 0 0; border:0;" | Take home messages |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * Further take home messages to be added to this feature box as the book chapter develops |} == See also == {{ic|Use bullet-points per Tutorial 2}} [[Achievement motivation]] (Book chapter 2011) [[wikipedia:Educational_psychology|Educational psychology]] (Wikipedia) [[wikipedia:Expectancy-value_theory|Expectancy-value theory]] (Wikipedia) [[wikipedia:Motivation|Motivation]] (Wikipedia) == External links == {{ic|Use bullet-points per Tutorial 2}} {{ic|Use alphabetical order}} [https://education.okstate.edu/site-files/documents/motivation-classrooms/motivation-minute-expectancy-value-theory.pdf Motivation minute: Expectancy-value theory] (Oklahoma State University, professional teaching resource) [https://www.simplypsychology.org/differences-between-extrinsic-and-intrinsic-motivation.html Extrinsic vs. Intrinsic Motivation: What’s the Difference?] (Verywell Mind article) ==References== {{Hanging indent|1= Bandura, A. (1997). Self-efficacy: The exercise of control. W. H. Freeman. Canning, E. A., Harackiewicz, J. M., Priniski, S. J., Hecht, C. A., Tibbetts, Y., & Hyde, J. S. (2018). Improving performance and retention in introductory biology with a utility-value intervention. Journal of Educational Psychology, 110(6), 834–849. https://doi.org/10.1037/edu0000244 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students' effort. Learning and Instruction, 47, 53–64. https://doi.org/10.1016/j.learninstruc.2016.10.009 Eccles, J. S. (2005). Subjective task value and the Eccles et al. model of achievement-related choices. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 105–121). Guilford Press. Eccles, J. S., Adler, T. F., Futterman, R., Goff, S. B., Kaczala, C. M., Meece, J. L., & Midgley, C. (1983). Expectancies, values, and academic behaviors. In J. T. Spence (Ed.), Achievement and achievement motives (pp. 75–146). W. H. Freeman. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eccles, J. S., & Wigfield, A. (2020). From expectancy–value theory to situated expectancy–value theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Flake, J. K., Barron, K. E., Hulleman, C., McCoach, B. D., & Welsh, M. E. (2015). Measuring cost: The forgotten component of expectancy-value theory. Contemporary Educational Psychology, 41, 232–244. https://doi.org/10.1016/j.cedpsych.2015.03.002 Gaspard, H., Dicke, A. L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Harackiewicz, J. M., & Priniski, S. J. (2018). Improving student outcomes in higher education: The science of targeted intervention. Annual Review of Psychology, 69, 409–435. https://doi.org/10.1146/annurev-psych-122216-011725 Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes. Science, 326(5958), 1410–1412. https://doi.org/10.1126/science.1177067 Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. https://doi.org/10.3102/00346543066004543 Perez, T., Cromley, J. G., & Kaplan, A. (2014). The role of identity development, values, and costs in college STEM retention. Journal of Educational Psychology, 106(1), 315–329. Pintrich, P. R. (2003). A Motivational Science Perspective on the Role of Student Motivation in Learning and Teaching Contexts. Journal of Educational Psychology, 95(4), 667–686. https://doi.org/10.1037/0022-0663.95.4.667 S00246164. (2025). Motivation Laptop.svg [SVG image]. Wikiversity. https://en.wikiversity.org/wiki/File:Motivation_Laptop.svg Schunk, D. H., & Pajares, F. (2009). Self-efficacy theory. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 35–53). Routledge. Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30, 1–35. https://doi.org/10.1016/j.dr.2009.12.001 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! Component !! Definition !! Example in education |- | '''Expectancy''' || Belief about ability to succeed || "I can pass this exam if I study." |- | '''Intrinsic value''' || Enjoyment or interest in the task || "I enjoy learning psychology." |- | '''Utility value''' || Usefulness for future goals || "This unit is required for my degree." |- | '''Attainment value''' || Importance to identity || "Being good at maths matters to me." |- | '''Cost''' || Negative aspects of engagement || "This assignment is stressful and time-consuming." |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> == Conclusion == *Arguably the most important section * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. ==External links== [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Education]] [[Category:Motivation and emotion/Book/Motivation]] mg7vhzkfo7c3nb78najaatyslqy8j7h 2834943 2834941 2026-09-29T01:11:19Z StudentUC2026 3105303 2834943 wikitext text/x-wiki {{title|Expectancy-value theory of educational motivation<br>''What is expectancy-value theory and how can it be applied to understand and enhance educational motivation?''}} __TOC__ ==Overview == {| style="width:100%; background:#FFE1F4; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#F9C3E5; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Imagine this scenario |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | <div style="background-color:#F6F4E7;{{Text color default}}; padding:8px; width:150px; float:right; text-align:center;"> [[File:Motivation Laptop.svg|150px|alt=A student reflecting on their motivation to learn]] <br> <span style="line-height:1.2; display:block; margin-top:2px;"> '''Figure 1'''. A student reflecting on their motivation to learn. </span> </div> You're sitting at your desk, working through a short maths quiz before dinner. The first question looks familiar, yet you pause, unsure whether you can solve it. You read it again, feeling a small wave of hesitation and mild frustration as you try to work out where to begin (see figure 1). After a moment, you recall solving a similar problem last week, and your teacher mentioned you were on the right track. This memory gives you a small boost of confidence, enough to try the first step. As you work through the question, you start thinking about why you’re even doing this quiz. You remind yourself that completing these tasks helps you prepare for the upcoming test and supports your longer‑term academic goals. This shift in mindset increases your confidence and clarifies the importance of the task. The quiz feels more manageable, and you start working through the first question. With this renewed sense of direction, you continue working through the quiz. This scenario demonstrates how expectancy and task value shape a learner’s motivation during routine academic activities. Expectancy-value theory provides a clear framework for understanding how changes in confidence and task value affect a learner’s motivation. This theory proposes that motivation increases when learners feel capable of succeeding and when they view a task as important, useful, or aligned with their goals. |} The scenario above shows how small shifts in confidence and perceived value can make everyday learning tasks feel either overwhelming or manageable. Many students experience fluctuating motivation during routine academic tasks, which can make learning feel overwhelming or discouraging. Expectancy-value theory helps explain this pattern by showing how confidence (expectancy beliefs) and perceived importance or usefulness (task values) shape a learner’s motivation. This theory provides a [[wikipedia:Social_cognition|social-cognitive]] explanation of why students choose to engage in learning activities and offers evidence-based strategies to strengthen both expectancy beliefs and task values. The following sections outline these core components and show how to apply them in educational settings to support more adaptive learning behaviours. {| style="width:100%; background:#CBF2D3; color:black; border-radius:12px; border:0;" |- ! style="background:#96E25B; color:black; border-radius:12px 12px 0 0; border:0;" | Focus questions |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * What is expectancy-value theory? * How do social‑cognitive models explain educational motivation? * How do expectancy beliefs influence educational motivation? * How do task values influence educational motivation? * How can expectancy-value theory be used to enhance motivation in educational settings? |} == Expectancy-value theory == [[File:Situated Expectancy-Value Theory SEVT Diagram.png|thumb|right|450px|Figure 2. Conceptual diagram illustrating expectancy‑value theory (EVT). Diagram created by author.]] * Introduce Eccles & Wigfield (2002) and explain that expectancy-value theory (EVT) was developed within achievement motivation research, focusing on how students’ expectations for success and the value they place on tasks influence their motivation and choices (see Figure 2). * Eccles and Wigfield (2000, 2002) describe expectancy-value theory as a major explanation of achievement motivation. The theory proposes that people are motivated when they believe they can succeed at a task and when they see the task as worthwhile. For example, a student is more likely to study for a maths test if they feel confident in their ability and if the test seems important, enjoyable, or useful. * Outline the key components of expectancy-value theory by explaining that motivation is shaped by two main factors: a person’s expectations for success and the value they attach to the task. Expectations reflect beliefs about one’s ability to perform well, while task value captures how important, enjoyable, or useful the activity feels. Together, these factors influence the choices students make and the effort they invest. * Although EVT is widely used in educational motivation research, scholars have noted several limitations. Pintrich (2003) argues that the theory places too much emphasis on rational decision-making and does not fully account for the emotional and social factors that influence student motivation. Other researchers highlight that EVT provides limited explanation of how expectancies and values develop over time or why they differ between students (Flake et al., 2015). These critiques suggest that EVT may not fully capture the complexity of real-world motivation. As shown in Table 1, expectancy-value theory consists of five core components that influence students’ motivation. '''Table 1.''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! style="background:#B3BBE4;{{Text color default}};" | Component ! style="background:#B3BBE4;{{Text color default}};" | Definition ! style="background:#B3BBE4;{{Text color default}};" | Example in education |- | style="background:#CEEEF4;{{Text color default}};" | '''Expectancy''' | style="background:#CEEEF4;{{Text color default}};" | Belief about ability to succeed | style="background:#CEEEF4;{{Text color default}};" | "I can pass this exam if I study." |- | style="background:#CEEEF4;;{{Text color default}};" | '''Intrinsic value''' | style="background:#CEEEF4;{{Text color default}};" | Enjoyment or interest in the task | style="background:#CEEEF4;{{Text color default}};" | "I enjoy learning psychology." |- | style="background:#CEEEF4;{{Text color default}};" | '''Utility value''' | style="background:#CEEEF4;{{Text color default}};" | Usefulness for future goals | style="background:#CEEEF4;{{Text color default}};" | "This unit is required for my degree." |- | style="background:#CEEEF4;{{Text color default}};" | '''Attainment value''' | style="background:#CEEEF4;{{Text color default}};" | Importance to identity | style="background:#CEEEF4;{{Text color default}};" | "Being good at maths matters to me." |- | style="background:#CEEEF4;{{Text color default}};" | '''Cost''' | style="background:#CEEEF4;{{Text color default}};" | Negative aspects of engagement | style="background:#CEEEF4;{{Text color default}};" | "This assignment is stressful and time-consuming." |} ==How do social-cognitive models explain educational motivation?== {{expand}} === Expectancy beliefs === * Expectancy beliefs refer to students’ judgments about how capable they are of successfully performing a task. In social cognitive models, these beliefs are central because they strongly predict academic choices, effort, and persistence. Eccles and Wigfield (2002) emphasise that expectancies for success are shaped by students’ prior achievement, self-concept, and perceptions of task difficulty. When students believe they can succeed, they are more likely to engage with learning activities and persist through challenges. * Introduce Bandura (1997), Pajares (1996), and Schunk and Pajares (2009) as key social cognitive researchers who demonstrate that expectancy beliefs strongly predict students’ effort, persistence, and academic performance. * Expectancy beliefs help explain educational motivation because students who believe they can succeed are more likely to engage with learning tasks, persist through challenges, and choose activities that support academic growth. This shows how social cognitive models account for differences in motivation across students with similar abilities. * Expectancy beliefs have been critiqued for focusing heavily on cognitive judgements and underestimating emotional and contextual influences on motivation. Research shows that confidence alone does not fully explain student engagement, particularly when anxiety, stress, or classroom climate reduce their willingness to participate (Pajares, 1996; Bandura, 1997). === Task value === * Introduce Eccles and Wigfield (2000, 2002) by explaining that task values describe how important, enjoyable, or useful students perceive a learning activity to be. * Eccles and Wigfield (2000, 2002) explain that task values include intrinsic value (enjoyment), utility value (usefulness), attainment value (importance), and cost, which refers to the perceived negative aspects of engaging in a task. These values shape how worthwhile students believe a learning activity is. * Research shows that students are more motivated when they find a task enjoyable or personally meaningful (Eccles & Wigfield, 2000). Utility value also increases motivation, with studies demonstrating that students work harder when they believe a task will support their future goals (Hulleman & Harackiewicz, 2009). * More recent work highlights the importance of cost, showing that high perceived effort, anxiety, or time demands can reduce motivation even when other values are high (Flake et al., 2015). This demonstrates that negative perceptions can undermine engagement despite strong intrinsic or utility value. * Task values help explain educational motivation because students are more likely to engage with learning activities, they find enjoyable, important, or useful. This shows how social cognitive models account for differences in motivation by focusing on how students evaluate the worth of academic tasks. * Task value research has been critiqued for focusing mainly on cognitive evaluations and overlooking how emotions, social pressures, and classroom climate shape students’ perceptions of value. Researchers also note that the different types of value can blur together in real learning situations, which makes it challenging to distinguish them clearly (Flake et al., 2015; Eccles, 2005{{ic|Use alphabetical order}}). Table #. Table caption goes here {| style="width:100%; background:#DBEAFA; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#B2E4FF; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | What's happening in this space? |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | '''The renaissance of cost'''<br> While early EVT research focused heavily on confidence and utility, contemporary studies increasingly emphasise the cost component. Recent work shows that psychological strain, performance anxiety, and time investment, captured across multidimensional cost categories such as effort, emotional strain, and opportunity costs, can meaningfully undermine high task values in demanding subjects like mathematics (Flake et al., 2015; Perez et al., 2014; Gaspard et al., 2015).<br> '''Real time and dynamic motivation'''<br> Motivation is now often viewed as a shifting, context-sensitive process rather than a static mindset. Recent studies show that students’ values and expectancies vary across lessons and classroom environments, influenced by factors such as perceived support, peer interactions, and task characteristics, which aligns with EVT’s emphasis on how beliefs and values develop within specific learning contexts (Eccles & Wigfield, 2020; Lazarides & Raufelder, 2017; Dietrich et al., 2017).<br> '''Utility value interventions'''<br> Rather than a static mindset, academic motivation is increasingly understood as a dynamic, context-sensitive process. Grounded in contemporary situated expectancy-value theory (SEVT), empirical evidence demonstrates that students’ expectancies for success and task values show meaningful micro-fluctuations across days, lessons, and distinct learning spaces (Eccles & Wigfield, 2020; Dietrich et al., 2017). These real-time shifts are shaped by immediate environmental inputs, including classroom climate, peer dynamics, and specific task designs (Lazarides & Raufelder, 2017).<br> '''The move towards situated expectancy-value theory'''<br> Early EVT (Eccles et al., 1983) focused on stable beliefs about competence and task value. Later refinements clarified how social influences (teachers, parents, and peers) shape these beliefs. SEVT (2020) reframes motivation as dynamic and situated, showing that expectancies and values shift in response to classroom context, feedback, identity, and social interactions. |} == How can expectancy-value theory be used to enhance motivation in educational settings? == * Strengthening expectancy beliefs: EVT can enhance motivation by increasing students’ expectancy beliefs through clear feedback, modelling effective strategies, and providing structured support that builds confidence in their ability to succeed (Bandura, 1997; Schunk & Pajares, 2009). * Enhancing task values: Teachers can increase task values by emphasising the relevance, usefulness, and personal meaning of learning activities, helping students connect academic tasks to future goals and real‑world applications (Eccles & Wigfield, 2002; Hulleman & Harackiewicz, 2009). * Reducing perceived cost: EVT can be applied by reducing perceived cost, such as anxiety, workload pressure, or time demands, through supportive classroom climates, manageable task structures, and opportunities for autonomy (Flake et al., 2015). * A practical example of EVT in educational settings is the utility‑value intervention, where students reflect on how a learning task connects to their personal goals or future plans. This approach has been shown to boost motivation and academic performance, particularly for students with lower initial expectancy beliefs (Hulleman & Harackiewicz, 2009). * A critique of applying EVT in classrooms is that increasing task values or expectancy beliefs may not be effective if students face structural barriers such as poor teaching quality, limited resources, or high stress. EVT interventions can also be difficult to sustain over time without consistent support (Eccles & Wigfield, 2002). == Knowledge checkpoint == {| style="width:50%; background:#F4CECE; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#E07F87; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | <big>Quiz</big> |- | style="border-radius:0 0 12px 12px; padding:2px; border:0;" | <div style="margin-top:-15px;"> <quiz display=simple> {Which component of expectancy-value theory refers to a student's belief about their ability to succeed? |type="()"} + Expectancy - Intrinsic value - Utility value - Cost {Which value component {{ic|of what?}} reflects how enjoyable or interesting a task is? |type="()"} - Attainment value + Intrinsic value - Utility value - Cost </quiz> </div> |} ==Conclusion== • Expectancy beliefs play a central role in educational motivation by shaping students’ confidence in their ability to succeed, influencing effort, persistence, and engagement (Eccles & Wigfield, 2002). • Task values determine how important, interesting, or useful a learning activity feels, guiding students’ willingness to invest time and energy in academic tasks (Wigfield & Eccles, 2000). • Expectancy-value theory shows that motivation increases when students believe they can succeed and when they view tasks as meaningful, relevant, or aligned with personal goals (Eccles & Wigfield, 2020). • Applying EVT in educational settings involves strengthening both expectancy beliefs and task values through supportive feedback, meaningful learning activities, and reduced perceived cost (Hulleman & Harackiewicz, 2009; Flake et al., 2015). • Together, these components provide a practical framework for understanding and improving student motivation in everyday classroom contexts (Eccles & Wigfield, 2002). {| style="width:100%; background:#FEFB9F; color:black; border-radius:12px; border:0;" |- ! style="background:#FFD208; color:black; border-radius:12px 12px 0 0; border:0;" | Take home messages |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * Further take home messages to be added to this feature box as the book chapter develops |} == See also == {{ic|Use bullet-points per Tutorial 2}} [[Achievement motivation]] (Book chapter 2011) [[wikipedia:Educational_psychology|Educational psychology]] (Wikipedia) [[wikipedia:Expectancy-value_theory|Expectancy-value theory]] (Wikipedia) [[wikipedia:Motivation|Motivation]] (Wikipedia) == External links == {{ic|Use bullet-points per Tutorial 2}} {{ic|Use alphabetical order}} [https://education.okstate.edu/site-files/documents/motivation-classrooms/motivation-minute-expectancy-value-theory.pdf Motivation minute: Expectancy-value theory] (Oklahoma State University, professional teaching resource) [https://www.simplypsychology.org/differences-between-extrinsic-and-intrinsic-motivation.html Extrinsic vs. Intrinsic Motivation: What’s the Difference?] (Verywell Mind article) ==References== {{Hanging indent|1= Bandura, A. (1997). Self-efficacy: The exercise of control. W. H. Freeman. Canning, E. A., Harackiewicz, J. M., Priniski, S. J., Hecht, C. A., Tibbetts, Y., & Hyde, J. S. (2018). Improving performance and retention in introductory biology with a utility-value intervention. Journal of Educational Psychology, 110(6), 834–849. https://doi.org/10.1037/edu0000244 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students' effort. Learning and Instruction, 47, 53–64. https://doi.org/10.1016/j.learninstruc.2016.10.009 Eccles, J. S. (2005). Subjective task value and the Eccles et al. model of achievement-related choices. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 105–121). Guilford Press. Eccles, J. S., Adler, T. F., Futterman, R., Goff, S. B., Kaczala, C. M., Meece, J. L., & Midgley, C. (1983). Expectancies, values, and academic behaviors. In J. T. Spence (Ed.), Achievement and achievement motives (pp. 75–146). W. H. Freeman. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eccles, J. S., & Wigfield, A. (2020). From expectancy–value theory to situated expectancy–value theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Flake, J. K., Barron, K. E., Hulleman, C., McCoach, B. D., & Welsh, M. E. (2015). Measuring cost: The forgotten component of expectancy-value theory. Contemporary Educational Psychology, 41, 232–244. https://doi.org/10.1016/j.cedpsych.2015.03.002 Gaspard, H., Dicke, A. L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Harackiewicz, J. M., & Priniski, S. J. (2018). Improving student outcomes in higher education: The science of targeted intervention. Annual Review of Psychology, 69, 409–435. https://doi.org/10.1146/annurev-psych-122216-011725 Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes. Science, 326(5958), 1410–1412. https://doi.org/10.1126/science.1177067 Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. https://doi.org/10.3102/00346543066004543 Perez, T., Cromley, J. G., & Kaplan, A. (2014). The role of identity development, values, and costs in college STEM retention. Journal of Educational Psychology, 106(1), 315–329. Pintrich, P. R. (2003). A Motivational Science Perspective on the Role of Student Motivation in Learning and Teaching Contexts. Journal of Educational Psychology, 95(4), 667–686. https://doi.org/10.1037/0022-0663.95.4.667 S00246164. (2025). Motivation Laptop.svg [SVG image]. Wikiversity. https://en.wikiversity.org/wiki/File:Motivation_Laptop.svg Schunk, D. H., & Pajares, F. (2009). Self-efficacy theory. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 35–53). Routledge. Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30, 1–35. https://doi.org/10.1016/j.dr.2009.12.001 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! Component !! Definition !! Example in education |- | '''Expectancy''' || Belief about ability to succeed || "I can pass this exam if I study." |- | '''Intrinsic value''' || Enjoyment or interest in the task || "I enjoy learning psychology." |- | '''Utility value''' || Usefulness for future goals || "This unit is required for my degree." |- | '''Attainment value''' || Importance to identity || "Being good at maths matters to me." |- | '''Cost''' || Negative aspects of engagement || "This assignment is stressful and time-consuming." |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> == Conclusion == *Arguably the most important section * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. ==External links== [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Education]] [[Category:Motivation and emotion/Book/Motivation]] 263ztqkb4hoybwwcv4vxu3621q4b4pf 2834944 2834943 2026-09-29T01:14:29Z StudentUC2026 3105303 2834944 wikitext text/x-wiki {{title|Expectancy-value theory of educational motivation<br>''What is expectancy-value theory and how can it be applied to understand and enhance educational motivation?''}} __TOC__ ==Overview == {| style="width:100%; background:#FFE1F4; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#F9C3E5; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Imagine this scenario |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | <div style="background-color:#F6F4E7;{{Text color default}}; padding:8px; width:150px; float:right; text-align:center;"> [[File:Motivation Laptop.svg|150px|alt=A student reflecting on their motivation to learn]] <br> <span style="line-height:1.2; display:block; margin-top:2px;"> '''Figure 1'''. A student reflecting on their motivation to learn. </span> </div> You're sitting at your desk, working through a short maths quiz before dinner. The first question looks familiar, yet you pause, unsure whether you can solve it. You read it again, feeling a small wave of hesitation and mild frustration as you try to work out where to begin (see figure 1). After a moment, you recall solving a similar problem last week, and your teacher mentioned you were on the right track. This memory gives you a small boost of confidence, enough to try the first step. As you work through the question, you start thinking about why you’re even doing this quiz. You remind yourself that completing these tasks helps you prepare for the upcoming test and supports your longer‑term academic goals. This shift in mindset increases your confidence and clarifies the importance of the task. The quiz feels more manageable, and you start working through the first question. With this renewed sense of direction, you continue working through the quiz. This scenario demonstrates how expectancy and task value shape a learner’s motivation during routine academic activities. Expectancy-value theory provides a clear framework for understanding how changes in confidence and task value affect a learner’s motivation. This theory proposes that motivation increases when learners feel capable of succeeding and when they view a task as important, useful, or aligned with their goals. |} The scenario above shows how small shifts in confidence and perceived value can make everyday learning tasks feel either overwhelming or manageable. Many students experience fluctuating motivation during routine academic tasks, which can make learning feel overwhelming or discouraging. Expectancy-value theory helps explain this pattern by showing how confidence (expectancy beliefs) and perceived importance or usefulness (task values) shape a learner’s motivation. This theory provides a [[wikipedia:Social_cognition|social-cognitive]] explanation of why students choose to engage in learning activities and offers evidence-based strategies to strengthen both expectancy beliefs and task values. The following sections outline these core components and show how to apply them in educational settings to support more adaptive learning behaviours. {| style="width:100%; background:#CBF2D3; color:black; border-radius:12px; border:0;" |- ! style="background:#96E25B; color:black; border-radius:12px 12px 0 0; border:0;" | Focus questions |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * What is expectancy-value theory? * How do social‑cognitive models explain educational motivation? * How do expectancy beliefs influence educational motivation? * How do task values influence educational motivation? * How can expectancy-value theory be used to enhance motivation in educational settings? |} == Expectancy-value theory == [[File:Situated Expectancy-Value Theory SEVT Diagram.png|thumb|right|450px|Figure 2. Conceptual diagram illustrating expectancy‑value theory (EVT). Diagram created by author.]] * Introduce Eccles & Wigfield (2002) and explain that expectancy-value theory (EVT) was developed within achievement motivation research, focusing on how students’ expectations for success and the value they place on tasks influence their motivation and choices (see Figure 2). * Eccles and Wigfield (2000, 2002) describe expectancy-value theory as a major explanation of achievement motivation. The theory proposes that people are motivated when they believe they can succeed at a task and when they see the task as worthwhile. For example, a student is more likely to study for a maths test if they feel confident in their ability and if the test seems important, enjoyable, or useful. * Outline the key components of expectancy-value theory by explaining that motivation is shaped by two main factors: a person’s expectations for success and the value they attach to the task. Expectations reflect beliefs about one’s ability to perform well, while task value captures how important, enjoyable, or useful the activity feels. Together, these factors influence the choices students make and the effort they invest. * Although EVT is widely used in educational motivation research, scholars have noted several limitations. Pintrich (2003) argues that the theory places too much emphasis on rational decision-making and does not fully account for the emotional and social factors that influence student motivation. Other researchers highlight that EVT provides limited explanation of how expectancies and values develop over time or why they differ between students (Flake et al., 2015). These critiques suggest that EVT may not fully capture the complexity of real-world motivation. As shown in Table 1, expectancy-value theory consists of five core components that influence students’ motivation. '''Table 1.''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! style="background:#B3BBE4;{{Text color default}};" | Component ! style="background:#B3BBE4;{{Text color default}};" | Definition ! style="background:#B3BBE4;{{Text color default}};" | Example in education |- | style="background:#CEEEF4;{{Text color default}};" | '''Expectancy''' | style="background:#CEEEF4;{{Text color default}};" | Belief about ability to succeed | style="background:#CEEEF4;{{Text color default}};" | "I can pass this exam if I study." |- | style="background:#CEEEF4;;{{Text color default}};" | '''Intrinsic value''' | style="background:#CEEEF4;{{Text color default}};" | Enjoyment or interest in the task | style="background:#CEEEF4;{{Text color default}};" | "I enjoy learning psychology." |- | style="background:#CEEEF4;{{Text color default}};" | '''Utility value''' | style="background:#CEEEF4;{{Text color default}};" | Usefulness for future goals | style="background:#CEEEF4;{{Text color default}};" | "This unit is required for my degree." |- | style="background:#CEEEF4;{{Text color default}};" | '''Attainment value''' | style="background:#CEEEF4;{{Text color default}};" | Importance to identity | style="background:#CEEEF4;{{Text color default}};" | "Being good at maths matters to me." |- | style="background:#CEEEF4;{{Text color default}};" | '''Cost''' | style="background:#CEEEF4;{{Text color default}};" | Negative aspects of engagement | style="background:#CEEEF4;{{Text color default}};" | "This assignment is stressful and time-consuming." |} ==How do social-cognitive models explain educational motivation?== {{expand}} === Expectancy beliefs === * Expectancy beliefs refer to students’ judgments about how capable they are of successfully performing a task. In social cognitive models, these beliefs are central because they strongly predict academic choices, effort, and persistence. Eccles and Wigfield (2002) emphasise that expectancies for success are shaped by students’ prior achievement, self-concept, and perceptions of task difficulty. When students believe they can succeed, they are more likely to engage with learning activities and persist through challenges. * Introduce Bandura (1997), Pajares (1996), and Schunk and Pajares (2009) as key social cognitive researchers who demonstrate that expectancy beliefs strongly predict students’ effort, persistence, and academic performance. * Expectancy beliefs help explain educational motivation because students who believe they can succeed are more likely to engage with learning tasks, persist through challenges, and choose activities that support academic growth. This shows how social cognitive models account for differences in motivation across students with similar abilities. * Expectancy beliefs have been critiqued for focusing heavily on cognitive judgements and underestimating emotional and contextual influences on motivation. Research shows that confidence alone does not fully explain student engagement, particularly when anxiety, stress, or classroom climate reduce their willingness to participate (Pajares, 1996; Bandura, 1997). === Task value === * Introduce Eccles and Wigfield (2000, 2002) by explaining that task values describe how important, enjoyable, or useful students perceive a learning activity to be. * Eccles and Wigfield (2000, 2002) explain that task values include intrinsic value (enjoyment), utility value (usefulness), attainment value (importance), and cost, which refers to the perceived negative aspects of engaging in a task. These values shape how worthwhile students believe a learning activity is. * Research shows that students are more motivated when they find a task enjoyable or personally meaningful (Eccles & Wigfield, 2000). Utility value also increases motivation, with studies demonstrating that students work harder when they believe a task will support their future goals (Hulleman & Harackiewicz, 2009). * More recent work highlights the importance of cost, showing that high perceived effort, anxiety, or time demands can reduce motivation even when other values are high (Flake et al., 2015). This demonstrates that negative perceptions can undermine engagement despite strong intrinsic or utility value. * Task values help explain educational motivation because students are more likely to engage with learning activities, they find enjoyable, important, or useful. This shows how social cognitive models account for differences in motivation by focusing on how students evaluate the worth of academic tasks. * Task value research has been critiqued for focusing mainly on cognitive evaluations and overlooking how emotions, social pressures, and classroom climate shape students’ perceptions of value. Researchers also note that the different types of value can blur together in real learning situations, which makes it challenging to distinguish them clearly (Flake et al., 2015; Eccles, 2005{{ic|Use alphabetical order}}). Table #. Table caption goes here {| style="width:100%; background:#DBEAFA; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#B2E4FF; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | What's happening in this space? |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | '''The renaissance of cost'''<br> While early EVT research focused heavily on confidence and utility, contemporary studies increasingly emphasise the cost component. Recent work shows that psychological strain, performance anxiety, and time investment, captured across multidimensional cost categories such as effort, emotional strain, and opportunity costs, can meaningfully undermine high task values in demanding subjects like mathematics (Flake et al., 2015; Perez et al., 2014; Gaspard et al., 2015).<br> '''Real-time and dynamic motivation'''<br> Motivation is now often viewed as a shifting, context-sensitive process rather than a static mindset. Recent studies show that students’ values and expectancies vary across lessons and classroom environments, influenced by factors such as perceived support, peer interactions, and task characteristics, which aligns with EVT’s emphasis on how beliefs and values develop within specific learning contexts (Eccles & Wigfield, 2020; Lazarides & Raufelder, 2017; Dietrich et al., 2017).<br> '''Utility value interventions'''<br> Rather than a static mindset, academic motivation is increasingly understood as a dynamic, context-sensitive process. Grounded in contemporary situated expectancy-value theory (SEVT), empirical evidence demonstrates that students’ expectancies for success and task values show meaningful micro-fluctuations across days, lessons, and distinct learning spaces (Eccles & Wigfield, 2020; Dietrich et al., 2017). These real-time shifts are shaped by immediate environmental inputs, including classroom climate, peer dynamics, and specific task designs (Lazarides & Raufelder, 2017).<br> '''The move towards situated expectancy-value theory'''<br> Early EVT (Eccles et al., 1983) focused on stable beliefs about competence and task value. Later refinements clarified how social influences (teachers, parents, and peers) shape these beliefs. SEVT (2020) reframes motivation as dynamic and situated, showing that expectancies and values shift in response to classroom context, feedback, identity, and social interactions. |} == How can expectancy-value theory be used to enhance motivation in educational settings? == * Strengthening expectancy beliefs: EVT can enhance motivation by increasing students’ expectancy beliefs through clear feedback, modelling effective strategies, and providing structured support that builds confidence in their ability to succeed (Bandura, 1997; Schunk & Pajares, 2009). * Enhancing task values: Teachers can increase task values by emphasising the relevance, usefulness, and personal meaning of learning activities, helping students connect academic tasks to future goals and real‑world applications (Eccles & Wigfield, 2002; Hulleman & Harackiewicz, 2009). * Reducing perceived cost: EVT can be applied by reducing perceived cost, such as anxiety, workload pressure, or time demands, through supportive classroom climates, manageable task structures, and opportunities for autonomy (Flake et al., 2015). * A practical example of EVT in educational settings is the utility‑value intervention, where students reflect on how a learning task connects to their personal goals or future plans. This approach has been shown to boost motivation and academic performance, particularly for students with lower initial expectancy beliefs (Hulleman & Harackiewicz, 2009). * A critique of applying EVT in classrooms is that increasing task values or expectancy beliefs may not be effective if students face structural barriers such as poor teaching quality, limited resources, or high stress. EVT interventions can also be difficult to sustain over time without consistent support (Eccles & Wigfield, 2002). == Knowledge checkpoint == {| style="width:50%; background:#F4CECE; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#E07F87; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | <big>Quiz</big> |- | style="border-radius:0 0 12px 12px; padding:2px; border:0;" | <div style="margin-top:-15px;"> <quiz display=simple> {Which component of expectancy-value theory refers to a student's belief about their ability to succeed? |type="()"} + Expectancy - Intrinsic value - Utility value - Cost {Which value component {{ic|of what?}} reflects how enjoyable or interesting a task is? |type="()"} - Attainment value + Intrinsic value - Utility value - Cost </quiz> </div> |} ==Conclusion== • Expectancy beliefs play a central role in educational motivation by shaping students’ confidence in their ability to succeed, influencing effort, persistence, and engagement (Eccles & Wigfield, 2002). • Task values determine how important, interesting, or useful a learning activity feels, guiding students’ willingness to invest time and energy in academic tasks (Wigfield & Eccles, 2000). • Expectancy-value theory shows that motivation increases when students believe they can succeed and when they view tasks as meaningful, relevant, or aligned with personal goals (Eccles & Wigfield, 2020). • Applying EVT in educational settings involves strengthening both expectancy beliefs and task values through supportive feedback, meaningful learning activities, and reduced perceived cost (Hulleman & Harackiewicz, 2009; Flake et al., 2015). • Together, these components provide a practical framework for understanding and improving student motivation in everyday classroom contexts (Eccles & Wigfield, 2002). {| style="width:100%; background:#FEFB9F; color:black; border-radius:12px; border:0;" |- ! style="background:#FFD208; color:black; border-radius:12px 12px 0 0; border:0;" | Take-home messages |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * Further take home messages to be added to this feature box as the book chapter develops |} == See also == {{ic|Use bullet-points per Tutorial 2}} [[Achievement motivation]] (Book chapter 2011) [[wikipedia:Educational_psychology|Educational psychology]] (Wikipedia) [[wikipedia:Expectancy-value_theory|Expectancy-value theory]] (Wikipedia) [[wikipedia:Motivation|Motivation]] (Wikipedia) == External links == {{ic|Use bullet-points per Tutorial 2}} {{ic|Use alphabetical order}} [https://education.okstate.edu/site-files/documents/motivation-classrooms/motivation-minute-expectancy-value-theory.pdf Motivation minute: Expectancy-value theory] (Oklahoma State University, professional teaching resource) [https://www.simplypsychology.org/differences-between-extrinsic-and-intrinsic-motivation.html Extrinsic vs. Intrinsic Motivation: What’s the Difference?] (Verywell Mind article) ==References== {{Hanging indent|1= Bandura, A. (1997). Self-efficacy: The exercise of control. W. H. Freeman. Canning, E. A., Harackiewicz, J. M., Priniski, S. J., Hecht, C. A., Tibbetts, Y., & Hyde, J. S. (2018). Improving performance and retention in introductory biology with a utility-value intervention. Journal of Educational Psychology, 110(6), 834–849. https://doi.org/10.1037/edu0000244 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students' effort. Learning and Instruction, 47, 53–64. https://doi.org/10.1016/j.learninstruc.2016.10.009 Eccles, J. S. (2005). Subjective task value and the Eccles et al. model of achievement-related choices. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 105–121). Guilford Press. Eccles, J. S., Adler, T. F., Futterman, R., Goff, S. B., Kaczala, C. M., Meece, J. L., & Midgley, C. (1983). Expectancies, values, and academic behaviors. In J. T. Spence (Ed.), Achievement and achievement motives (pp. 75–146). W. H. Freeman. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eccles, J. S., & Wigfield, A. (2020). From expectancy–value theory to situated expectancy–value theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Flake, J. K., Barron, K. E., Hulleman, C., McCoach, B. D., & Welsh, M. E. (2015). Measuring cost: The forgotten component of expectancy-value theory. Contemporary Educational Psychology, 41, 232–244. https://doi.org/10.1016/j.cedpsych.2015.03.002 Gaspard, H., Dicke, A. L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Harackiewicz, J. M., & Priniski, S. J. (2018). Improving student outcomes in higher education: The science of targeted intervention. Annual Review of Psychology, 69, 409–435. https://doi.org/10.1146/annurev-psych-122216-011725 Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes. Science, 326(5958), 1410–1412. https://doi.org/10.1126/science.1177067 Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. https://doi.org/10.3102/00346543066004543 Perez, T., Cromley, J. G., & Kaplan, A. (2014). The role of identity development, values, and costs in college STEM retention. Journal of Educational Psychology, 106(1), 315–329. Pintrich, P. R. (2003). A Motivational Science Perspective on the Role of Student Motivation in Learning and Teaching Contexts. Journal of Educational Psychology, 95(4), 667–686. https://doi.org/10.1037/0022-0663.95.4.667 S00246164. (2025). Motivation Laptop.svg [SVG image]. Wikiversity. https://en.wikiversity.org/wiki/File:Motivation_Laptop.svg Schunk, D. H., & Pajares, F. (2009). Self-efficacy theory. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 35–53). Routledge. Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30, 1–35. https://doi.org/10.1016/j.dr.2009.12.001 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! Component !! Definition !! Example in education |- | '''Expectancy''' || Belief about ability to succeed || "I can pass this exam if I study." |- | '''Intrinsic value''' || Enjoyment or interest in the task || "I enjoy learning psychology." |- | '''Utility value''' || Usefulness for future goals || "This unit is required for my degree." |- | '''Attainment value''' || Importance to identity || "Being good at maths matters to me." |- | '''Cost''' || Negative aspects of engagement || "This assignment is stressful and time-consuming." |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> == Conclusion == *Arguably the most important section * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. ==External links== [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Education]] [[Category:Motivation and emotion/Book/Motivation]] oxasmxk8fndac5s7solzss7o4ybmw3j 2834945 2834944 2026-09-29T01:16:33Z StudentUC2026 3105303 /* Expectancy-value theory */ 2834945 wikitext text/x-wiki {{title|Expectancy-value theory of educational motivation<br>''What is expectancy-value theory and how can it be applied to understand and enhance educational motivation?''}} __TOC__ ==Overview == {| style="width:100%; background:#FFE1F4; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#F9C3E5; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Imagine this scenario |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | <div style="background-color:#F6F4E7;{{Text color default}}; padding:8px; width:150px; float:right; text-align:center;"> [[File:Motivation Laptop.svg|150px|alt=A student reflecting on their motivation to learn]] <br> <span style="line-height:1.2; display:block; margin-top:2px;"> '''Figure 1'''. A student reflecting on their motivation to learn. </span> </div> You're sitting at your desk, working through a short maths quiz before dinner. The first question looks familiar, yet you pause, unsure whether you can solve it. You read it again, feeling a small wave of hesitation and mild frustration as you try to work out where to begin (see figure 1). After a moment, you recall solving a similar problem last week, and your teacher mentioned you were on the right track. This memory gives you a small boost of confidence, enough to try the first step. As you work through the question, you start thinking about why you’re even doing this quiz. You remind yourself that completing these tasks helps you prepare for the upcoming test and supports your longer‑term academic goals. This shift in mindset increases your confidence and clarifies the importance of the task. The quiz feels more manageable, and you start working through the first question. With this renewed sense of direction, you continue working through the quiz. This scenario demonstrates how expectancy and task value shape a learner’s motivation during routine academic activities. Expectancy-value theory provides a clear framework for understanding how changes in confidence and task value affect a learner’s motivation. This theory proposes that motivation increases when learners feel capable of succeeding and when they view a task as important, useful, or aligned with their goals. |} The scenario above shows how small shifts in confidence and perceived value can make everyday learning tasks feel either overwhelming or manageable. Many students experience fluctuating motivation during routine academic tasks, which can make learning feel overwhelming or discouraging. Expectancy-value theory helps explain this pattern by showing how confidence (expectancy beliefs) and perceived importance or usefulness (task values) shape a learner’s motivation. This theory provides a [[wikipedia:Social_cognition|social-cognitive]] explanation of why students choose to engage in learning activities and offers evidence-based strategies to strengthen both expectancy beliefs and task values. The following sections outline these core components and show how to apply them in educational settings to support more adaptive learning behaviours. {| style="width:100%; background:#CBF2D3; color:black; border-radius:12px; border:0;" |- ! style="background:#96E25B; color:black; border-radius:12px 12px 0 0; border:0;" | Focus questions |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * What is expectancy-value theory? * How do social‑cognitive models explain educational motivation? * How do expectancy beliefs influence educational motivation? * How do task values influence educational motivation? * How can expectancy-value theory be used to enhance motivation in educational settings? |} == What is expectancy-value theory? == [[File:Situated Expectancy-Value Theory SEVT Diagram.png|thumb|right|450px|Figure 2. Conceptual diagram illustrating expectancy‑value theory (EVT). Diagram created by author.]] * Introduce Eccles & Wigfield (2002) and explain that expectancy-value theory (EVT) was developed within achievement motivation research, focusing on how students’ expectations for success and the value they place on tasks influence their motivation and choices (see Figure 2). * Eccles and Wigfield (2000, 2002) describe expectancy-value theory as a major explanation of achievement motivation. The theory proposes that people are motivated when they believe they can succeed at a task and when they see the task as worthwhile. For example, a student is more likely to study for a maths test if they feel confident in their ability and if the test seems important, enjoyable, or useful. * Outline the key components of expectancy-value theory by explaining that motivation is shaped by two main factors: a person’s expectations for success and the value they attach to the task. Expectations reflect beliefs about one’s ability to perform well, while task value captures how important, enjoyable, or useful the activity feels. Together, these factors influence the choices students make and the effort they invest. * Although EVT is widely used in educational motivation research, scholars have noted several limitations. Pintrich (2003) argues that the theory places too much emphasis on rational decision-making and does not fully account for the emotional and social factors that influence student motivation. Other researchers highlight that EVT provides limited explanation of how expectancies and values develop over time or why they differ between students (Flake et al., 2015). These critiques suggest that EVT may not fully capture the complexity of real-world motivation. As shown in Table 1, expectancy-value theory consists of five core components that influence students’ motivation. '''Table 1.''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! style="background:#B3BBE4;{{Text color default}};" | Component ! style="background:#B3BBE4;{{Text color default}};" | Definition ! style="background:#B3BBE4;{{Text color default}};" | Example in education |- | style="background:#CEEEF4;{{Text color default}};" | '''Expectancy''' | style="background:#CEEEF4;{{Text color default}};" | Belief about ability to succeed | style="background:#CEEEF4;{{Text color default}};" | "I can pass this exam if I study." |- | style="background:#CEEEF4;;{{Text color default}};" | '''Intrinsic value''' | style="background:#CEEEF4;{{Text color default}};" | Enjoyment or interest in the task | style="background:#CEEEF4;{{Text color default}};" | "I enjoy learning psychology." |- | style="background:#CEEEF4;{{Text color default}};" | '''Utility value''' | style="background:#CEEEF4;{{Text color default}};" | Usefulness for future goals | style="background:#CEEEF4;{{Text color default}};" | "This unit is required for my degree." |- | style="background:#CEEEF4;{{Text color default}};" | '''Attainment value''' | style="background:#CEEEF4;{{Text color default}};" | Importance to identity | style="background:#CEEEF4;{{Text color default}};" | "Being good at maths matters to me." |- | style="background:#CEEEF4;{{Text color default}};" | '''Cost''' | style="background:#CEEEF4;{{Text color default}};" | Negative aspects of engagement | style="background:#CEEEF4;{{Text color default}};" | "This assignment is stressful and time-consuming." |} ==How do social-cognitive models explain educational motivation?== {{expand}} === Expectancy beliefs === * Expectancy beliefs refer to students’ judgments about how capable they are of successfully performing a task. In social cognitive models, these beliefs are central because they strongly predict academic choices, effort, and persistence. Eccles and Wigfield (2002) emphasise that expectancies for success are shaped by students’ prior achievement, self-concept, and perceptions of task difficulty. When students believe they can succeed, they are more likely to engage with learning activities and persist through challenges. * Introduce Bandura (1997), Pajares (1996), and Schunk and Pajares (2009) as key social cognitive researchers who demonstrate that expectancy beliefs strongly predict students’ effort, persistence, and academic performance. * Expectancy beliefs help explain educational motivation because students who believe they can succeed are more likely to engage with learning tasks, persist through challenges, and choose activities that support academic growth. This shows how social cognitive models account for differences in motivation across students with similar abilities. * Expectancy beliefs have been critiqued for focusing heavily on cognitive judgements and underestimating emotional and contextual influences on motivation. Research shows that confidence alone does not fully explain student engagement, particularly when anxiety, stress, or classroom climate reduce their willingness to participate (Pajares, 1996; Bandura, 1997). === Task value === * Introduce Eccles and Wigfield (2000, 2002) by explaining that task values describe how important, enjoyable, or useful students perceive a learning activity to be. * Eccles and Wigfield (2000, 2002) explain that task values include intrinsic value (enjoyment), utility value (usefulness), attainment value (importance), and cost, which refers to the perceived negative aspects of engaging in a task. These values shape how worthwhile students believe a learning activity is. * Research shows that students are more motivated when they find a task enjoyable or personally meaningful (Eccles & Wigfield, 2000). Utility value also increases motivation, with studies demonstrating that students work harder when they believe a task will support their future goals (Hulleman & Harackiewicz, 2009). * More recent work highlights the importance of cost, showing that high perceived effort, anxiety, or time demands can reduce motivation even when other values are high (Flake et al., 2015). This demonstrates that negative perceptions can undermine engagement despite strong intrinsic or utility value. * Task values help explain educational motivation because students are more likely to engage with learning activities, they find enjoyable, important, or useful. This shows how social cognitive models account for differences in motivation by focusing on how students evaluate the worth of academic tasks. * Task value research has been critiqued for focusing mainly on cognitive evaluations and overlooking how emotions, social pressures, and classroom climate shape students’ perceptions of value. Researchers also note that the different types of value can blur together in real learning situations, which makes it challenging to distinguish them clearly (Flake et al., 2015; Eccles, 2005{{ic|Use alphabetical order}}). Table #. Table caption goes here {| style="width:100%; background:#DBEAFA; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#B2E4FF; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | What's happening in this space? |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | '''The renaissance of cost'''<br> While early EVT research focused heavily on confidence and utility, contemporary studies increasingly emphasise the cost component. Recent work shows that psychological strain, performance anxiety, and time investment, captured across multidimensional cost categories such as effort, emotional strain, and opportunity costs, can meaningfully undermine high task values in demanding subjects like mathematics (Flake et al., 2015; Perez et al., 2014; Gaspard et al., 2015).<br> '''Real-time and dynamic motivation'''<br> Motivation is now often viewed as a shifting, context-sensitive process rather than a static mindset. Recent studies show that students’ values and expectancies vary across lessons and classroom environments, influenced by factors such as perceived support, peer interactions, and task characteristics, which aligns with EVT’s emphasis on how beliefs and values develop within specific learning contexts (Eccles & Wigfield, 2020; Lazarides & Raufelder, 2017; Dietrich et al., 2017).<br> '''Utility value interventions'''<br> Rather than a static mindset, academic motivation is increasingly understood as a dynamic, context-sensitive process. Grounded in contemporary situated expectancy-value theory (SEVT), empirical evidence demonstrates that students’ expectancies for success and task values show meaningful micro-fluctuations across days, lessons, and distinct learning spaces (Eccles & Wigfield, 2020; Dietrich et al., 2017). These real-time shifts are shaped by immediate environmental inputs, including classroom climate, peer dynamics, and specific task designs (Lazarides & Raufelder, 2017).<br> '''The move towards situated expectancy-value theory'''<br> Early EVT (Eccles et al., 1983) focused on stable beliefs about competence and task value. Later refinements clarified how social influences (teachers, parents, and peers) shape these beliefs. SEVT (2020) reframes motivation as dynamic and situated, showing that expectancies and values shift in response to classroom context, feedback, identity, and social interactions. |} == How can expectancy-value theory be used to enhance motivation in educational settings? == * Strengthening expectancy beliefs: EVT can enhance motivation by increasing students’ expectancy beliefs through clear feedback, modelling effective strategies, and providing structured support that builds confidence in their ability to succeed (Bandura, 1997; Schunk & Pajares, 2009). * Enhancing task values: Teachers can increase task values by emphasising the relevance, usefulness, and personal meaning of learning activities, helping students connect academic tasks to future goals and real‑world applications (Eccles & Wigfield, 2002; Hulleman & Harackiewicz, 2009). * Reducing perceived cost: EVT can be applied by reducing perceived cost, such as anxiety, workload pressure, or time demands, through supportive classroom climates, manageable task structures, and opportunities for autonomy (Flake et al., 2015). * A practical example of EVT in educational settings is the utility‑value intervention, where students reflect on how a learning task connects to their personal goals or future plans. This approach has been shown to boost motivation and academic performance, particularly for students with lower initial expectancy beliefs (Hulleman & Harackiewicz, 2009). * A critique of applying EVT in classrooms is that increasing task values or expectancy beliefs may not be effective if students face structural barriers such as poor teaching quality, limited resources, or high stress. EVT interventions can also be difficult to sustain over time without consistent support (Eccles & Wigfield, 2002). == Knowledge checkpoint == {| style="width:50%; background:#F4CECE; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#E07F87; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | <big>Quiz</big> |- | style="border-radius:0 0 12px 12px; padding:2px; border:0;" | <div style="margin-top:-15px;"> <quiz display=simple> {Which component of expectancy-value theory refers to a student's belief about their ability to succeed? |type="()"} + Expectancy - Intrinsic value - Utility value - Cost {Which value component {{ic|of what?}} reflects how enjoyable or interesting a task is? |type="()"} - Attainment value + Intrinsic value - Utility value - Cost </quiz> </div> |} ==Conclusion== • Expectancy beliefs play a central role in educational motivation by shaping students’ confidence in their ability to succeed, influencing effort, persistence, and engagement (Eccles & Wigfield, 2002). • Task values determine how important, interesting, or useful a learning activity feels, guiding students’ willingness to invest time and energy in academic tasks (Wigfield & Eccles, 2000). • Expectancy-value theory shows that motivation increases when students believe they can succeed and when they view tasks as meaningful, relevant, or aligned with personal goals (Eccles & Wigfield, 2020). • Applying EVT in educational settings involves strengthening both expectancy beliefs and task values through supportive feedback, meaningful learning activities, and reduced perceived cost (Hulleman & Harackiewicz, 2009; Flake et al., 2015). • Together, these components provide a practical framework for understanding and improving student motivation in everyday classroom contexts (Eccles & Wigfield, 2002). {| style="width:100%; background:#FEFB9F; color:black; border-radius:12px; border:0;" |- ! style="background:#FFD208; color:black; border-radius:12px 12px 0 0; border:0;" | Take-home messages |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * Further take home messages to be added to this feature box as the book chapter develops |} == See also == {{ic|Use bullet-points per Tutorial 2}} [[Achievement motivation]] (Book chapter 2011) [[wikipedia:Educational_psychology|Educational psychology]] (Wikipedia) [[wikipedia:Expectancy-value_theory|Expectancy-value theory]] (Wikipedia) [[wikipedia:Motivation|Motivation]] (Wikipedia) == External links == {{ic|Use bullet-points per Tutorial 2}} {{ic|Use alphabetical order}} [https://education.okstate.edu/site-files/documents/motivation-classrooms/motivation-minute-expectancy-value-theory.pdf Motivation minute: Expectancy-value theory] (Oklahoma State University, professional teaching resource) [https://www.simplypsychology.org/differences-between-extrinsic-and-intrinsic-motivation.html Extrinsic vs. Intrinsic Motivation: What’s the Difference?] (Verywell Mind article) ==References== {{Hanging indent|1= Bandura, A. (1997). Self-efficacy: The exercise of control. W. H. Freeman. Canning, E. A., Harackiewicz, J. M., Priniski, S. J., Hecht, C. A., Tibbetts, Y., & Hyde, J. S. (2018). Improving performance and retention in introductory biology with a utility-value intervention. Journal of Educational Psychology, 110(6), 834–849. https://doi.org/10.1037/edu0000244 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students' effort. Learning and Instruction, 47, 53–64. https://doi.org/10.1016/j.learninstruc.2016.10.009 Eccles, J. S. (2005). Subjective task value and the Eccles et al. model of achievement-related choices. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 105–121). Guilford Press. Eccles, J. S., Adler, T. F., Futterman, R., Goff, S. B., Kaczala, C. M., Meece, J. L., & Midgley, C. (1983). Expectancies, values, and academic behaviors. In J. T. Spence (Ed.), Achievement and achievement motives (pp. 75–146). W. H. Freeman. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eccles, J. S., & Wigfield, A. (2020). From expectancy–value theory to situated expectancy–value theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Flake, J. K., Barron, K. E., Hulleman, C., McCoach, B. D., & Welsh, M. E. (2015). Measuring cost: The forgotten component of expectancy-value theory. Contemporary Educational Psychology, 41, 232–244. https://doi.org/10.1016/j.cedpsych.2015.03.002 Gaspard, H., Dicke, A. L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Harackiewicz, J. M., & Priniski, S. J. (2018). Improving student outcomes in higher education: The science of targeted intervention. Annual Review of Psychology, 69, 409–435. https://doi.org/10.1146/annurev-psych-122216-011725 Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes. Science, 326(5958), 1410–1412. https://doi.org/10.1126/science.1177067 Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. https://doi.org/10.3102/00346543066004543 Perez, T., Cromley, J. G., & Kaplan, A. (2014). The role of identity development, values, and costs in college STEM retention. Journal of Educational Psychology, 106(1), 315–329. Pintrich, P. R. (2003). A Motivational Science Perspective on the Role of Student Motivation in Learning and Teaching Contexts. Journal of Educational Psychology, 95(4), 667–686. https://doi.org/10.1037/0022-0663.95.4.667 S00246164. (2025). Motivation Laptop.svg [SVG image]. Wikiversity. https://en.wikiversity.org/wiki/File:Motivation_Laptop.svg Schunk, D. H., & Pajares, F. (2009). Self-efficacy theory. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 35–53). Routledge. Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30, 1–35. https://doi.org/10.1016/j.dr.2009.12.001 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! Component !! Definition !! Example in education |- | '''Expectancy''' || Belief about ability to succeed || "I can pass this exam if I study." |- | '''Intrinsic value''' || Enjoyment or interest in the task || "I enjoy learning psychology." |- | '''Utility value''' || Usefulness for future goals || "This unit is required for my degree." |- | '''Attainment value''' || Importance to identity || "Being good at maths matters to me." |- | '''Cost''' || Negative aspects of engagement || "This assignment is stressful and time-consuming." |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> == Conclusion == *Arguably the most important section * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. ==External links== [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Education]] [[Category:Motivation and emotion/Book/Motivation]] gby81ncecaw922ex7w5sso22ifpibxi 2834947 2834945 2026-09-29T01:19:44Z StudentUC2026 3105303 2834947 wikitext text/x-wiki {{title|Expectancy-value theory of educational motivation<br>''What is expectancy-value theory and how can it be applied to understand and enhance educational motivation?''}} __TOC__ ==Overview == {| style="width:100%; background:#FFE1F4; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#F9C3E5; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Imagine this scenario |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | <div style="background-color:#F6F4E7;{{Text color default}}; padding:8px; width:150px; float:right; text-align:center;"> [[File:Motivation Laptop.svg|150px|alt=A student reflecting on their motivation to learn]] <br> <span style="line-height:1.2; display:block; margin-top:2px;"> '''Figure 1'''. A student reflecting on their motivation to learn. </span> </div> You're sitting at your desk, working through a short maths quiz before dinner. The first question looks familiar, yet you pause, unsure whether you can solve it. You read it again, feeling a small wave of hesitation and mild frustration as you try to work out where to begin (see figure 1). After a moment, you recall solving a similar problem last week, and your teacher mentioned you were on the right track. This memory gives you a small boost of confidence, enough to try the first step. As you work through the question, you start thinking about why you’re even doing this quiz. You remind yourself that completing these tasks helps you prepare for the upcoming test and supports your longer‑term academic goals. This shift in mindset increases your confidence and clarifies the importance of the task. The quiz feels more manageable, and you start working through the first question. With this renewed sense of direction, you continue working through the quiz. This scenario demonstrates how expectancy and task value shape a learner’s motivation during routine academic activities. Expectancy-value theory provides a clear framework for understanding how changes in confidence and task value affect a learner’s motivation. This theory proposes that motivation increases when learners feel capable of succeeding and when they view a task as important, useful, or aligned with their goals. |} The scenario above shows how small shifts in confidence and perceived value can make everyday learning tasks feel either overwhelming or manageable. Many students experience fluctuating motivation during routine academic tasks, which can make learning feel overwhelming or discouraging. Expectancy-value theory helps explain this pattern by showing how confidence (expectancy beliefs) and perceived importance or usefulness (task values) shape a learner’s motivation. This theory provides a [[wikipedia:Social_cognition|social-cognitive]] explanation of why students choose to engage in learning activities and offers evidence-based strategies to strengthen both expectancy beliefs and task values. The following sections outline these core components and show how to apply them in educational settings to support more adaptive learning behaviours. {| style="width:100%; background:#CBF2D3; color:black; border-radius:12px; border:0;" |- ! style="background:#96E25B; color:black; border-radius:12px 12px 0 0; border:0;" | Focus questions |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * What is expectancy-value theory? * How do social‑cognitive models explain educational motivation? * How do expectancy beliefs influence educational motivation? * How do task values influence educational motivation? * How can expectancy-value theory be used to enhance motivation in educational settings? |} == What is expectancy-value theory? == [[File:Situated Expectancy-Value Theory SEVT Diagram.png|thumb|right|450px|Figure 2. Conceptual diagram illustrating expectancy‑value theory (EVT). Diagram created by author.]] * Introduce Eccles & Wigfield (2002) and explain that expectancy-value theory (EVT) was developed within achievement motivation research, focusing on how students’ expectations for success and the value they place on tasks influence their motivation and choices (see Figure 2). * Eccles and Wigfield (2000, 2002) describe expectancy-value theory as a major explanation of achievement motivation. The theory proposes that people are motivated when they believe they can succeed at a task and when they see the task as worthwhile. For example, a student is more likely to study for a maths test if they feel confident in their ability and if the test seems important, enjoyable, or useful. * Outline the key components of expectancy-value theory by explaining that motivation is shaped by two main factors: a person’s expectations for success and the value they attach to the task. Expectations reflect beliefs about one’s ability to perform well, while task value captures how important, enjoyable, or useful the activity feels. Together, these factors influence the choices students make and the effort they invest. * Although EVT is widely used in educational motivation research, scholars have noted several limitations. Pintrich (2003) argues that the theory places too much emphasis on rational decision-making and does not fully account for the emotional and social factors that influence student motivation. Other researchers highlight that EVT provides limited explanation of how expectancies and values develop over time or why they differ between students (Flake et al., 2015). These critiques suggest that EVT may not fully capture the complexity of real-world motivation. As shown in Table 1, expectancy-value theory consists of five core components that influence students’ motivation. '''Table 1.''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! style="background:#B3BBE4;{{Text color default}};" | Component ! style="background:#B3BBE4;{{Text color default}};" | Definition ! style="background:#B3BBE4;{{Text color default}};" | Example in education |- | style="background:#CEEEF4;{{Text color default}};" | '''Expectancy''' | style="background:#CEEEF4;{{Text color default}};" | Belief about ability to succeed | style="background:#CEEEF4;{{Text color default}};" | "I can pass this exam if I study." |- | style="background:#CEEEF4;;{{Text color default}};" | '''Intrinsic value''' | style="background:#CEEEF4;{{Text color default}};" | Enjoyment or interest in the task | style="background:#CEEEF4;{{Text color default}};" | "I enjoy learning psychology." |- | style="background:#CEEEF4;{{Text color default}};" | '''Utility value''' | style="background:#CEEEF4;{{Text color default}};" | Usefulness for future goals | style="background:#CEEEF4;{{Text color default}};" | "This unit is required for my degree." |- | style="background:#CEEEF4;{{Text color default}};" | '''Attainment value''' | style="background:#CEEEF4;{{Text color default}};" | Importance to identity | style="background:#CEEEF4;{{Text color default}};" | "Being good at maths matters to me." |- | style="background:#CEEEF4;{{Text color default}};" | '''Cost''' | style="background:#CEEEF4;{{Text color default}};" | Negative aspects of engagement | style="background:#CEEEF4;{{Text color default}};" | "This assignment is stressful and time-consuming." |} ==How do social-cognitive models explain educational motivation?== {{expand}} === How do expectancy beliefs influence educational motivation? === * Expectancy beliefs refer to students’ judgments about how capable they are of successfully performing a task. In social cognitive models, these beliefs are central because they strongly predict academic choices, effort, and persistence. Eccles and Wigfield (2002) emphasise that expectancies for success are shaped by students’ prior achievement, self-concept, and perceptions of task difficulty. When students believe they can succeed, they are more likely to engage with learning activities and persist through challenges. * Introduce Bandura (1997), Pajares (1996), and Schunk and Pajares (2009) as key social cognitive researchers who demonstrate that expectancy beliefs strongly predict students’ effort, persistence, and academic performance. * Expectancy beliefs help explain educational motivation because students who believe they can succeed are more likely to engage with learning tasks, persist through challenges, and choose activities that support academic growth. This shows how social cognitive models account for differences in motivation across students with similar abilities. * Expectancy beliefs have been critiqued for focusing heavily on cognitive judgements and underestimating emotional and contextual influences on motivation. Research shows that confidence alone does not fully explain student engagement, particularly when anxiety, stress, or classroom climate reduce their willingness to participate (Pajares, 1996; Bandura, 1997). === How do task values influence educational motivation? === * Introduce Eccles and Wigfield (2000, 2002) by explaining that task values describe how important, enjoyable, or useful students perceive a learning activity to be. * Eccles and Wigfield (2000, 2002) explain that task values include intrinsic value (enjoyment), utility value (usefulness), attainment value (importance), and cost, which refers to the perceived negative aspects of engaging in a task. These values shape how worthwhile students believe a learning activity is. * Research shows that students are more motivated when they find a task enjoyable or personally meaningful (Eccles & Wigfield, 2000). Utility value also increases motivation, with studies demonstrating that students work harder when they believe a task will support their future goals (Hulleman & Harackiewicz, 2009). * More recent work highlights the importance of cost, showing that high perceived effort, anxiety, or time demands can reduce motivation even when other values are high (Flake et al., 2015). This demonstrates that negative perceptions can undermine engagement despite strong intrinsic or utility value. * Task values help explain educational motivation because students are more likely to engage with learning activities, they find enjoyable, important, or useful. This shows how social cognitive models account for differences in motivation by focusing on how students evaluate the worth of academic tasks. * Task value research has been critiqued for focusing mainly on cognitive evaluations and overlooking how emotions, social pressures, and classroom climate shape students’ perceptions of value. Researchers also note that the different types of value can blur together in real learning situations, which makes it challenging to distinguish them clearly (Flake et al., 2015; Eccles, 2005{{ic|Use alphabetical order}}). Table #. Table caption goes here {| style="width:100%; background:#DBEAFA; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#B2E4FF; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | What's happening in this space? |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | '''The renaissance of cost'''<br> While early EVT research focused heavily on confidence and utility, contemporary studies increasingly emphasise the cost component. Recent work shows that psychological strain, performance anxiety, and time investment, captured across multidimensional cost categories such as effort, emotional strain, and opportunity costs, can meaningfully undermine high task values in demanding subjects like mathematics (Flake et al., 2015; Perez et al., 2014; Gaspard et al., 2015).<br> '''Real-time and dynamic motivation'''<br> Motivation is now often viewed as a shifting, context-sensitive process rather than a static mindset. Recent studies show that students’ values and expectancies vary across lessons and classroom environments, influenced by factors such as perceived support, peer interactions, and task characteristics, which aligns with EVT’s emphasis on how beliefs and values develop within specific learning contexts (Eccles & Wigfield, 2020; Lazarides & Raufelder, 2017; Dietrich et al., 2017).<br> '''Utility value interventions'''<br> Rather than a static mindset, academic motivation is increasingly understood as a dynamic, context-sensitive process. Grounded in contemporary situated expectancy-value theory (SEVT), empirical evidence demonstrates that students’ expectancies for success and task values show meaningful micro-fluctuations across days, lessons, and distinct learning spaces (Eccles & Wigfield, 2020; Dietrich et al., 2017). These real-time shifts are shaped by immediate environmental inputs, including classroom climate, peer dynamics, and specific task designs (Lazarides & Raufelder, 2017).<br> '''The move towards situated expectancy-value theory'''<br> Early EVT (Eccles et al., 1983) focused on stable beliefs about competence and task value. Later refinements clarified how social influences (teachers, parents, and peers) shape these beliefs. SEVT (2020) reframes motivation as dynamic and situated, showing that expectancies and values shift in response to classroom context, feedback, identity, and social interactions. |} == How can expectancy-value theory be used to enhance motivation in educational settings? == * Strengthening expectancy beliefs: EVT can enhance motivation by increasing students’ expectancy beliefs through clear feedback, modelling effective strategies, and providing structured support that builds confidence in their ability to succeed (Bandura, 1997; Schunk & Pajares, 2009). * Enhancing task values: Teachers can increase task values by emphasising the relevance, usefulness, and personal meaning of learning activities, helping students connect academic tasks to future goals and real‑world applications (Eccles & Wigfield, 2002; Hulleman & Harackiewicz, 2009). * Reducing perceived cost: EVT can be applied by reducing perceived cost, such as anxiety, workload pressure, or time demands, through supportive classroom climates, manageable task structures, and opportunities for autonomy (Flake et al., 2015). * A practical example of EVT in educational settings is the utility‑value intervention, where students reflect on how a learning task connects to their personal goals or future plans. This approach has been shown to boost motivation and academic performance, particularly for students with lower initial expectancy beliefs (Hulleman & Harackiewicz, 2009). * A critique of applying EVT in classrooms is that increasing task values or expectancy beliefs may not be effective if students face structural barriers such as poor teaching quality, limited resources, or high stress. EVT interventions can also be difficult to sustain over time without consistent support (Eccles & Wigfield, 2002). == Knowledge checkpoint == {| style="width:50%; background:#F4CECE; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#E07F87; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | <big>Quiz</big> |- | style="border-radius:0 0 12px 12px; padding:2px; border:0;" | <div style="margin-top:-15px;"> <quiz display=simple> {Which component of expectancy-value theory refers to a student's belief about their ability to succeed? |type="()"} + Expectancy - Intrinsic value - Utility value - Cost {Which value component {{ic|of what?}} reflects how enjoyable or interesting a task is? |type="()"} - Attainment value + Intrinsic value - Utility value - Cost </quiz> </div> |} ==Conclusion== • Expectancy beliefs play a central role in educational motivation by shaping students’ confidence in their ability to succeed, influencing effort, persistence, and engagement (Eccles & Wigfield, 2002). • Task values determine how important, interesting, or useful a learning activity feels, guiding students’ willingness to invest time and energy in academic tasks (Wigfield & Eccles, 2000). • Expectancy-value theory shows that motivation increases when students believe they can succeed and when they view tasks as meaningful, relevant, or aligned with personal goals (Eccles & Wigfield, 2020). • Applying EVT in educational settings involves strengthening both expectancy beliefs and task values through supportive feedback, meaningful learning activities, and reduced perceived cost (Hulleman & Harackiewicz, 2009; Flake et al., 2015). • Together, these components provide a practical framework for understanding and improving student motivation in everyday classroom contexts (Eccles & Wigfield, 2002). {| style="width:100%; background:#FEFB9F; color:black; border-radius:12px; border:0;" |- ! style="background:#FFD208; color:black; border-radius:12px 12px 0 0; border:0;" | Take-home messages |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * Further take home messages to be added to this feature box as the book chapter develops |} == See also == {{ic|Use bullet-points per Tutorial 2}} [[Achievement motivation]] (Book chapter 2011) [[wikipedia:Educational_psychology|Educational psychology]] (Wikipedia) [[wikipedia:Expectancy-value_theory|Expectancy-value theory]] (Wikipedia) [[wikipedia:Motivation|Motivation]] (Wikipedia) == External links == {{ic|Use bullet-points per Tutorial 2}} {{ic|Use alphabetical order}} [https://education.okstate.edu/site-files/documents/motivation-classrooms/motivation-minute-expectancy-value-theory.pdf Motivation minute: Expectancy-value theory] (Oklahoma State University, professional teaching resource) [https://www.simplypsychology.org/differences-between-extrinsic-and-intrinsic-motivation.html Extrinsic vs. Intrinsic Motivation: What’s the Difference?] (Verywell Mind article) ==References== {{Hanging indent|1= Bandura, A. (1997). Self-efficacy: The exercise of control. W. H. Freeman. Canning, E. A., Harackiewicz, J. M., Priniski, S. J., Hecht, C. A., Tibbetts, Y., & Hyde, J. S. (2018). Improving performance and retention in introductory biology with a utility-value intervention. Journal of Educational Psychology, 110(6), 834–849. https://doi.org/10.1037/edu0000244 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students' effort. Learning and Instruction, 47, 53–64. https://doi.org/10.1016/j.learninstruc.2016.10.009 Eccles, J. S. (2005). Subjective task value and the Eccles et al. model of achievement-related choices. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 105–121). Guilford Press. Eccles, J. S., Adler, T. F., Futterman, R., Goff, S. B., Kaczala, C. M., Meece, J. L., & Midgley, C. (1983). Expectancies, values, and academic behaviors. In J. T. Spence (Ed.), Achievement and achievement motives (pp. 75–146). W. H. Freeman. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eccles, J. S., & Wigfield, A. (2020). From expectancy–value theory to situated expectancy–value theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Flake, J. K., Barron, K. E., Hulleman, C., McCoach, B. D., & Welsh, M. E. (2015). Measuring cost: The forgotten component of expectancy-value theory. Contemporary Educational Psychology, 41, 232–244. https://doi.org/10.1016/j.cedpsych.2015.03.002 Gaspard, H., Dicke, A. L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Harackiewicz, J. M., & Priniski, S. J. (2018). Improving student outcomes in higher education: The science of targeted intervention. Annual Review of Psychology, 69, 409–435. https://doi.org/10.1146/annurev-psych-122216-011725 Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes. Science, 326(5958), 1410–1412. https://doi.org/10.1126/science.1177067 Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. https://doi.org/10.3102/00346543066004543 Perez, T., Cromley, J. G., & Kaplan, A. (2014). The role of identity development, values, and costs in college STEM retention. Journal of Educational Psychology, 106(1), 315–329. Pintrich, P. R. (2003). A Motivational Science Perspective on the Role of Student Motivation in Learning and Teaching Contexts. Journal of Educational Psychology, 95(4), 667–686. https://doi.org/10.1037/0022-0663.95.4.667 S00246164. (2025). Motivation Laptop.svg [SVG image]. Wikiversity. https://en.wikiversity.org/wiki/File:Motivation_Laptop.svg Schunk, D. H., & Pajares, F. (2009). Self-efficacy theory. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 35–53). Routledge. Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30, 1–35. https://doi.org/10.1016/j.dr.2009.12.001 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! Component !! Definition !! Example in education |- | '''Expectancy''' || Belief about ability to succeed || "I can pass this exam if I study." |- | '''Intrinsic value''' || Enjoyment or interest in the task || "I enjoy learning psychology." |- | '''Utility value''' || Usefulness for future goals || "This unit is required for my degree." |- | '''Attainment value''' || Importance to identity || "Being good at maths matters to me." |- | '''Cost''' || Negative aspects of engagement || "This assignment is stressful and time-consuming." |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> == Conclusion == *Arguably the most important section * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. ==External links== [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Education]] [[Category:Motivation and emotion/Book/Motivation]] qxi3ws19wij6br0tp075wn7vebk5lkg 2834948 2834947 2026-09-29T01:36:00Z StudentUC2026 3105303 Added meta-analysis reference to reference list 2834948 wikitext text/x-wiki {{title|Expectancy-value theory of educational motivation<br>''What is expectancy-value theory and how can it be applied to understand and enhance educational motivation?''}} __TOC__ ==Overview == {| style="width:100%; background:#FFE1F4; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#F9C3E5; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | Imagine this scenario |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | <div style="background-color:#F6F4E7;{{Text color default}}; padding:8px; width:150px; float:right; text-align:center;"> [[File:Motivation Laptop.svg|150px|alt=A student reflecting on their motivation to learn]] <br> <span style="line-height:1.2; display:block; margin-top:2px;"> '''Figure 1'''. A student reflecting on their motivation to learn. </span> </div> You're sitting at your desk, working through a short maths quiz before dinner. The first question looks familiar, yet you pause, unsure whether you can solve it. You read it again, feeling a small wave of hesitation and mild frustration as you try to work out where to begin (see figure 1). After a moment, you recall solving a similar problem last week, and your teacher mentioned you were on the right track. This memory gives you a small boost of confidence, enough to try the first step. As you work through the question, you start thinking about why you’re even doing this quiz. You remind yourself that completing these tasks helps you prepare for the upcoming test and supports your longer‑term academic goals. This shift in mindset increases your confidence and clarifies the importance of the task. The quiz feels more manageable, and you start working through the first question. With this renewed sense of direction, you continue working through the quiz. This scenario demonstrates how expectancy and task value shape a learner’s motivation during routine academic activities. Expectancy-value theory provides a clear framework for understanding how changes in confidence and task value affect a learner’s motivation. This theory proposes that motivation increases when learners feel capable of succeeding and when they view a task as important, useful, or aligned with their goals. |} The scenario above shows how small shifts in confidence and perceived value can make everyday learning tasks feel either overwhelming or manageable. Many students experience fluctuating motivation during routine academic tasks, which can make learning feel overwhelming or discouraging. Expectancy-value theory helps explain this pattern by showing how confidence (expectancy beliefs) and perceived importance or usefulness (task values) shape a learner’s motivation. This theory provides a [[wikipedia:Social_cognition|social-cognitive]] explanation of why students choose to engage in learning activities and offers evidence-based strategies to strengthen both expectancy beliefs and task values. The following sections outline these core components and show how to apply them in educational settings to support more adaptive learning behaviours. {| style="width:100%; background:#CBF2D3; color:black; border-radius:12px; border:0;" |- ! style="background:#96E25B; color:black; border-radius:12px 12px 0 0; border:0;" | Focus questions |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * What is expectancy-value theory? * How do social‑cognitive models explain educational motivation? * How do expectancy beliefs influence educational motivation? * How do task values influence educational motivation? * How can expectancy-value theory be used to enhance motivation in educational settings? |} == What is expectancy-value theory? == [[File:Situated Expectancy-Value Theory SEVT Diagram.png|thumb|right|450px|Figure 2. Conceptual diagram illustrating expectancy‑value theory (EVT). Diagram created by author.]] * Introduce Eccles & Wigfield (2002) and explain that expectancy-value theory (EVT) was developed within achievement motivation research, focusing on how students’ expectations for success and the value they place on tasks influence their motivation and choices (see Figure 2). * Eccles and Wigfield (2000, 2002) describe expectancy-value theory as a major explanation of achievement motivation. The theory proposes that people are motivated when they believe they can succeed at a task and when they see the task as worthwhile. For example, a student is more likely to study for a maths test if they feel confident in their ability and if the test seems important, enjoyable, or useful. * Outline the key components of expectancy-value theory by explaining that motivation is shaped by two main factors: a person’s expectations for success and the value they attach to the task. Expectations reflect beliefs about one’s ability to perform well, while task value captures how important, enjoyable, or useful the activity feels. Together, these factors influence the choices students make and the effort they invest. * Although EVT is widely used in educational motivation research, scholars have noted several limitations. Pintrich (2003) argues that the theory places too much emphasis on rational decision-making and does not fully account for the emotional and social factors that influence student motivation. Other researchers highlight that EVT provides limited explanation of how expectancies and values develop over time or why they differ between students (Flake et al., 2015). These critiques suggest that EVT may not fully capture the complexity of real-world motivation. As shown in Table 1, expectancy-value theory consists of five core components that influence students’ motivation. '''Table 1.''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! style="background:#B3BBE4;{{Text color default}};" | Component ! style="background:#B3BBE4;{{Text color default}};" | Definition ! style="background:#B3BBE4;{{Text color default}};" | Example in education |- | style="background:#CEEEF4;{{Text color default}};" | '''Expectancy''' | style="background:#CEEEF4;{{Text color default}};" | Belief about ability to succeed | style="background:#CEEEF4;{{Text color default}};" | "I can pass this exam if I study." |- | style="background:#CEEEF4;;{{Text color default}};" | '''Intrinsic value''' | style="background:#CEEEF4;{{Text color default}};" | Enjoyment or interest in the task | style="background:#CEEEF4;{{Text color default}};" | "I enjoy learning psychology." |- | style="background:#CEEEF4;{{Text color default}};" | '''Utility value''' | style="background:#CEEEF4;{{Text color default}};" | Usefulness for future goals | style="background:#CEEEF4;{{Text color default}};" | "This unit is required for my degree." |- | style="background:#CEEEF4;{{Text color default}};" | '''Attainment value''' | style="background:#CEEEF4;{{Text color default}};" | Importance to identity | style="background:#CEEEF4;{{Text color default}};" | "Being good at maths matters to me." |- | style="background:#CEEEF4;{{Text color default}};" | '''Cost''' | style="background:#CEEEF4;{{Text color default}};" | Negative aspects of engagement | style="background:#CEEEF4;{{Text color default}};" | "This assignment is stressful and time-consuming." |} ==How do social-cognitive models explain educational motivation?== {{expand}} === How do expectancy beliefs influence educational motivation? === * Expectancy beliefs refer to students’ judgments about how capable they are of successfully performing a task. In social cognitive models, these beliefs are central because they strongly predict academic choices, effort, and persistence. Eccles and Wigfield (2002) emphasise that expectancies for success are shaped by students’ prior achievement, self-concept, and perceptions of task difficulty. When students believe they can succeed, they are more likely to engage with learning activities and persist through challenges. * Introduce Bandura (1997), Pajares (1996), and Schunk and Pajares (2009) as key social cognitive researchers who demonstrate that expectancy beliefs strongly predict students’ effort, persistence, and academic performance. * Expectancy beliefs help explain educational motivation because students who believe they can succeed are more likely to engage with learning tasks, persist through challenges, and choose activities that support academic growth. This shows how social cognitive models account for differences in motivation across students with similar abilities. * Expectancy beliefs have been critiqued for focusing heavily on cognitive judgements and underestimating emotional and contextual influences on motivation. Research shows that confidence alone does not fully explain student engagement, particularly when anxiety, stress, or classroom climate reduce their willingness to participate (Pajares, 1996; Bandura, 1997). === How do task values influence educational motivation? === * Introduce Eccles and Wigfield (2000, 2002) by explaining that task values describe how important, enjoyable, or useful students perceive a learning activity to be. * Eccles and Wigfield (2000, 2002) explain that task values include intrinsic value (enjoyment), utility value (usefulness), attainment value (importance), and cost, which refers to the perceived negative aspects of engaging in a task. These values shape how worthwhile students believe a learning activity is. * Research shows that students are more motivated when they find a task enjoyable or personally meaningful (Eccles & Wigfield, 2000). Utility value also increases motivation, with studies demonstrating that students work harder when they believe a task will support their future goals (Hulleman & Harackiewicz, 2009). * More recent work highlights the importance of cost, showing that high perceived effort, anxiety, or time demands can reduce motivation even when other values are high (Flake et al., 2015). This demonstrates that negative perceptions can undermine engagement despite strong intrinsic or utility value. * Task values help explain educational motivation because students are more likely to engage with learning activities, they find enjoyable, important, or useful. This shows how social cognitive models account for differences in motivation by focusing on how students evaluate the worth of academic tasks. * Task value research has been critiqued for focusing mainly on cognitive evaluations and overlooking how emotions, social pressures, and classroom climate shape students’ perceptions of value. Researchers also note that the different types of value can blur together in real learning situations, which makes it challenging to distinguish them clearly (Flake et al., 2015; Eccles, 2005{{ic|Use alphabetical order}}). Table #. Table caption goes here {| style="width:100%; background:#DBEAFA; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#B2E4FF; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | What's happening in this space? |- | style="border-radius:0 0 12px 12px; padding:8px; border:0;" | '''The renaissance of cost'''<br> While early EVT research focused heavily on confidence and utility, contemporary studies increasingly emphasise the cost component. Recent work shows that psychological strain, performance anxiety, and time investment, captured across multidimensional cost categories such as effort, emotional strain, and opportunity costs, can meaningfully undermine high task values in demanding subjects like mathematics (Flake et al., 2015; Perez et al., 2014; Gaspard et al., 2015).<br> '''Real-time and dynamic motivation'''<br> Motivation is now often viewed as a shifting, context-sensitive process rather than a static mindset. Recent studies show that students’ values and expectancies vary across lessons and classroom environments, influenced by factors such as perceived support, peer interactions, and task characteristics, which aligns with EVT’s emphasis on how beliefs and values develop within specific learning contexts (Eccles & Wigfield, 2020; Lazarides & Raufelder, 2017; Dietrich et al., 2017).<br> '''Utility value interventions'''<br> Rather than a static mindset, academic motivation is increasingly understood as a dynamic, context-sensitive process. Grounded in contemporary situated expectancy-value theory (SEVT), empirical evidence demonstrates that students’ expectancies for success and task values show meaningful micro-fluctuations across days, lessons, and distinct learning spaces (Eccles & Wigfield, 2020; Dietrich et al., 2017). These real-time shifts are shaped by immediate environmental inputs, including classroom climate, peer dynamics, and specific task designs (Lazarides & Raufelder, 2017).<br> '''The move towards situated expectancy-value theory'''<br> Early EVT (Eccles et al., 1983) focused on stable beliefs about competence and task value. Later refinements clarified how social influences (teachers, parents, and peers) shape these beliefs. SEVT (2020) reframes motivation as dynamic and situated, showing that expectancies and values shift in response to classroom context, feedback, identity, and social interactions. |} == How can expectancy-value theory be used to enhance motivation in educational settings? == * Strengthening expectancy beliefs: EVT can enhance motivation by increasing students’ expectancy beliefs through clear feedback, modelling effective strategies, and providing structured support that builds confidence in their ability to succeed (Bandura, 1997; Schunk & Pajares, 2009). * Enhancing task values: Teachers can increase task values by emphasising the relevance, usefulness, and personal meaning of learning activities, helping students connect academic tasks to future goals and real‑world applications (Eccles & Wigfield, 2002; Hulleman & Harackiewicz, 2009). * Reducing perceived cost: EVT can be applied by reducing perceived cost, such as anxiety, workload pressure, or time demands, through supportive classroom climates, manageable task structures, and opportunities for autonomy (Flake et al., 2015). * A practical example of EVT in educational settings is the utility‑value intervention, where students reflect on how a learning task connects to their personal goals or future plans. This approach has been shown to boost motivation and academic performance, particularly for students with lower initial expectancy beliefs (Hulleman & Harackiewicz, 2009). * A critique of applying EVT in classrooms is that increasing task values or expectancy beliefs may not be effective if students face structural barriers such as poor teaching quality, limited resources, or high stress. EVT interventions can also be difficult to sustain over time without consistent support (Eccles & Wigfield, 2002). == Knowledge checkpoint == {| style="width:50%; background:#F4CECE; color:black; border-radius:12px; border:0; padding:2px;" |- ! style="background:#E07F87; color:black; border-radius:12px 12px 0 0; border:0; padding:2px;" | <big>Quiz</big> |- | style="border-radius:0 0 12px 12px; padding:2px; border:0;" | <div style="margin-top:-15px;"> <quiz display=simple> {Which component of expectancy-value theory refers to a student's belief about their ability to succeed? |type="()"} + Expectancy - Intrinsic value - Utility value - Cost {Which value component {{ic|of what?}} reflects how enjoyable or interesting a task is? |type="()"} - Attainment value + Intrinsic value - Utility value - Cost </quiz> </div> |} ==Conclusion== • Expectancy beliefs play a central role in educational motivation by shaping students’ confidence in their ability to succeed, influencing effort, persistence, and engagement (Eccles & Wigfield, 2002). • Task values determine how important, interesting, or useful a learning activity feels, guiding students’ willingness to invest time and energy in academic tasks (Wigfield & Eccles, 2000). • Expectancy-value theory shows that motivation increases when students believe they can succeed and when they view tasks as meaningful, relevant, or aligned with personal goals (Eccles & Wigfield, 2020). • Applying EVT in educational settings involves strengthening both expectancy beliefs and task values through supportive feedback, meaningful learning activities, and reduced perceived cost (Hulleman & Harackiewicz, 2009; Flake et al., 2015). • Together, these components provide a practical framework for understanding and improving student motivation in everyday classroom contexts (Eccles & Wigfield, 2002). {| style="width:100%; background:#FEFB9F; color:black; border-radius:12px; border:0;" |- ! style="background:#FFD208; color:black; border-radius:12px 12px 0 0; border:0;" | Take-home messages |- | style="border-radius:0 0 12px 12px; padding:10px; border:0;" | * Further take home messages to be added to this feature box as the book chapter develops |} == See also == {{ic|Use bullet-points per Tutorial 2}} [[Achievement motivation]] (Book chapter 2011) [[wikipedia:Educational_psychology|Educational psychology]] (Wikipedia) [[wikipedia:Expectancy-value_theory|Expectancy-value theory]] (Wikipedia) [[wikipedia:Motivation|Motivation]] (Wikipedia) == External links == {{ic|Use bullet-points per Tutorial 2}} {{ic|Use alphabetical order}} [https://education.okstate.edu/site-files/documents/motivation-classrooms/motivation-minute-expectancy-value-theory.pdf Motivation minute: Expectancy-value theory] (Oklahoma State University, professional teaching resource) [https://www.simplypsychology.org/differences-between-extrinsic-and-intrinsic-motivation.html Extrinsic vs. Intrinsic Motivation: What’s the Difference?] (Verywell Mind article) ==References== {{Hanging indent|1= Bandura, A. (1997). Self-efficacy: The exercise of control. W. H. Freeman. Canning, E. A., Harackiewicz, J. M., Priniski, S. J., Hecht, C. A., Tibbetts, Y., & Hyde, J. S. (2018). Improving performance and retention in introductory biology with a utility-value intervention. Journal of Educational Psychology, 110(6), 834–849. https://doi.org/10.1037/edu0000244 Dietrich, J., Viljaranta, J., Moeller, J., & Kracke, B. (2017). Situational expectancies and task values: Associations with students' effort. Learning and Instruction, 47, 53–64. https://doi.org/10.1016/j.learninstruc.2016.10.009 Eccles, J. S. (2005). Subjective task value and the Eccles et al. model of achievement-related choices. In A. J. Elliot & C. S. Dweck (Eds.), Handbook of competence and motivation (pp. 105–121). Guilford Press. Eccles, J. S., Adler, T. F., Futterman, R., Goff, S. B., Kaczala, C. M., Meece, J. L., & Midgley, C. (1983). Expectancies, values, and academic behaviors. In J. T. Spence (Ed.), Achievement and achievement motives (pp. 75–146). W. H. Freeman. Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values, and goals. Annual Review of Psychology, 53(1), 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Eccles, J. S., & Wigfield, A. (2020). From expectancy–value theory to situated expectancy–value theory: A developmental, social cognitive, and sociocultural perspective on motivation. Contemporary Educational Psychology, 61, Article 101859. https://doi.org/10.1016/j.cedpsych.2020.101859 Flake, J. K., Barron, K. E., Hulleman, C., McCoach, B. D., & Welsh, M. E. (2015). Measuring cost: The forgotten component of expectancy-value theory. Contemporary Educational Psychology, 41, 232–244. https://doi.org/10.1016/j.cedpsych.2015.03.002 Gaspard, H., Dicke, A. L., Flunger, B., Schreier, B., Häfner, I., Trautwein, U., & Nagengast, B. (2015). More value through greater differentiation: Gender differences in value beliefs about math. Journal of Educational Psychology, 107(3), 663–677. https://doi.org/10.1037/edu0000003 Harackiewicz, J. M., & Priniski, S. J. (2018). Improving student outcomes in higher education: The science of targeted intervention. Annual Review of Psychology, 69, 409–435. https://doi.org/10.1146/annurev-psych-122216-011725 Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting interest and performance in high school science classes. Science, 326(5958), 1410–1412. https://doi.org/10.1126/science.1177067 Lazowski, R. A., & Hulleman, C. S. (2016). Motivation Interventions in Education: A Meta-Analytic Review. Review of Educational Research, 86(2), 602–640. https://doi.org/10.3102/0034654315617832 Pajares, F. (1996). Self-efficacy beliefs in academic settings. Review of Educational Research, 66(4), 543–578. https://doi.org/10.3102/00346543066004543 Perez, T., Cromley, J. G., & Kaplan, A. (2014). The role of identity development, values, and costs in college STEM retention. Journal of Educational Psychology, 106(1), 315–329. Pintrich, P. R. (2003). A Motivational Science Perspective on the Role of Student Motivation in Learning and Teaching Contexts. Journal of Educational Psychology, 95(4), 667–686. https://doi.org/10.1037/0022-0663.95.4.667 S00246164. (2025). Motivation Laptop.svg [SVG image]. Wikiversity. https://en.wikiversity.org/wiki/File:Motivation_Laptop.svg Schunk, D. H., & Pajares, F. (2009). Self-efficacy theory. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of motivation at school (pp. 35–53). Routledge. Wigfield, A., & Cambria, J. (2010). Students’ achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes. Developmental Review, 30, 1–35. https://doi.org/10.1016/j.dr.2009.12.001 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' Key components of expectancy-value theory and their relevance to educational motivation. {| class="wikitable" style="margin: auto;" |- ! Component !! Definition !! Example in education |- | '''Expectancy''' || Belief about ability to succeed || "I can pass this exam if I study." |- | '''Intrinsic value''' || Enjoyment or interest in the task || "I enjoy learning psychology." |- | '''Utility value''' || Usefulness for future goals || "This unit is required for my degree." |- | '''Attainment value''' || Importance to identity || "Being good at maths matters to me." |- | '''Cost''' || Negative aspects of engagement || "This assignment is stressful and time-consuming." |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> == Conclusion == *Arguably the most important section * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. ==External links== [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Education]] [[Category:Motivation and emotion/Book/Motivation]] gkeqroat3zh64jfh8b3ufvcevk4psd9 0 331234 2834845 2834844 2026-09-28T12:09:59Z Mymunu 3106327 /* Working memory and cognitive process */ 2834845 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" [[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play.]] |'''Case study: Part 1''' A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, child enjoys choosing where and what to play and moving between the different activities. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and avoids the activity. After watching for short time, the child returns, suggests for new idea to continue game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation theory provides one way of understanding on how children may respond to emotional situations during play (see figure 1) (Gross, 2015). |} Outdoor play can involve more than physical movement. It is more than simply being outside and engaging in activities across playgrounds, gardens, streets and natural spaces. Outdoor play involves exploration, social interaction, choice and challenge and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). These experiences may create emotional situations in which children need to respond to frustration, disagreement, uncertainty or excitement [''add a reference'']. Gross's (2015) [[emotional regulation]] theory provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Experiences are shaped by the different opportunities provided by different physical and social environments, which means the same outdoor space may not provide the same experience for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's emotional well-being. {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and school support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' === Children's emotional well-being === Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social and emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess in different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How does social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. * Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. * Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The play ground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult. Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal contributions can affect who child are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] acihaxbbqv3ptoudyhxqdyj2x4bkkrt 2834918 2834845 2026-09-28T22:02:05Z Mymunu 3106327 Overview 2834918 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and school support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Visualise outdoor play as providing children with opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play activity [''add a reference'']. It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [[emotional regulation]], social-emotional competence, positive and negative affect, and internalising or externalising symptoms. These differences are significant because outdoor play and emotional well-being are not measured as single, consistent constructs across research. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' === Children's emotional well-being === Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social and emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess in different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How does social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. * Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. * Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The play ground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult. Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal contributions can affect who child are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] 714jraf9uvc2y6sq08zzvk72xrpiuhz 2834922 2834918 2026-09-28T22:23:56Z Mymunu 3106327 /* Understanding outdoor play and children's emotional well-being */ 2834922 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|265x265px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity which occurs in outdoor environments like gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, which means children to freely interact with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as same as outdoor physical activity. Davenport et al., (2025) examined outdoor play separately from outdoor moderate- to- vigorous physical activity (MVPA). This distinction is significant because research on children's outdoor play is not necessarily measuring the same behaviour as research on the intensity of physical activity. [''this paragraph is fairly repetitive of the one above, are you able to condense to minimise repetition?]'' === Children's emotional well-being === Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social and emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess in different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How does social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. * Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. * Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The play ground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult. Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal contributions can affect who child are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] 0dn3kkwnmgejll5qsvb65csmzylmtl2 2834926 2834922 2026-09-28T22:40:30Z Mymunu 3106327 /* Outdoor play */ 2834926 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have autonomy to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to- vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing and outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social and emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess in different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How does social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. * Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. * Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The play ground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult. Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal contributions can affect who child are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] mgroy6j0281wuch7ua480r85guenenj 2834929 2834926 2026-09-28T22:48:55Z Mymunu 3106327 /* Children's emotional well-being */ 2834929 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when examining whether outdoor play is associated with children's emotional well-being. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|177x177px|'''Figure 3'''. ''Children may experience different emotional responses during social outdoor play'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. When research findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How does social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. * Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. * Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The play ground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult. Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal contributions can affect who child are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] cujr4vgeqcupdhsk1xkia2lg3z8ja1b 2834967 2834929 2026-09-29T05:33:14Z Mymunu 3106327 /* Children's emotional well-being */ 2834967 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the case study part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes. Possible pathways discussed in the recent research include [[Emotional regulation|emotion regulation]], working memory, social interaction, physical activity, connection with the nature and adventurous play. However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play literatures remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological response is one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotional regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how a children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Furthermore, research on play provides some association between outdoor play and emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to situations that occur during outdoor play.'']]When the rule of game change, the child becomes frustrated and decided to step away. After a short time the child returns and suggests for different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al., (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysfunction. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al., (2025) reported this as full mediation. This suggests working memory may be involved in the association between afternoon play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How does social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on the opportunities available in children's social and physical environment. Children may have access to same outdoor space but may not have same opportunities in play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied the affordance theory to children's outdoor play and examined outdoor play transactions and development of an environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The study examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The study also differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. Based on Pereira et al. (2024) and Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has climbing feature, sand pit area and open natural space. * Kim quickly joins a group of children in a climbing frame. The climbing frame structure provides him with an opportunity for physical play and social interactions. * Leo uses sand pit area with other child, he plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. Both the examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The play ground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communication and play together, while joining unfamiliar children or existing game could be difficult. Affective social competence provides a psychological framework for understanding these social interactions (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) described emotional competence as involving experiencing, sending and receiving affect, together with awareness, identification, responding with the social context and regulation. These process may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play (Gal-Szabo et al., 2018). Gal-Szabo et al. (2018) also found that situational understanding of the emotions was related to uninvolved behaviour and was marginally related to more social play. However, that the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === Autonomy can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environment can support or restrict children's participation (Beekhuizen et al., 2025). Their identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusion about the effects of individual outdoor play space features.. Gibson et al. (2017) reported a similar limitation for loose-parts play. Their systematic review found insufficient quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish a particular environmental feature that improves children's emotional well-being ( Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo uses the same school playground. Kim already knows many children and easily joins the group activity. Leo uses a mobility device and finds some areas of playground is difficult to access. Leo also does not know many children who are already playing. Leo also prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate is different. Affordance theory can help to understand how the physical environment provides opportunities depending on child and how the space can be used. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal contributions can affect who child are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] kc9izotr7xklcye8gkuck82d2nqgjwf 0 331243 2834859 2834678 2026-09-28T15:28:14Z JSINST 3110286 2834859 wikitext text/x-wiki ------------- '''Design of the project''' (Informed Consent Form below) This is an international, multicentre, observational longitudinal health database that uses routinely collected maritime health examination data to monitor the prevalence and trends of prediabetes, diabetes T-2, hypertension, prehypertension, overweight, obesity and established prevention among maritime workers and students. The intention is to register and offer standardized prevention Inspired by the “Danish health database” (ref. 1)References 1. Health of the Danes https://www.danskernessundhed.dk/ '''Strengths of the design''' • Uses routinely collected data, making it relatively inexpensive. • Can accumulate large international datasets. • Reflects real-world clinical practice. • Supports long-term surveillance of maritime workers' health. • Allow for evaluation of preventive interventions within routine healthcare. Organisers• The John Snow Prediabetes Institute in collaboration with: • Maritime Medical Health Clinics, • The International Maritime Health Associtation (IMHA), • the Filippine Maritime Health Association '''Objectives''' • To establish an “International maritime health database” with samples of health examination data from the maritime clinics and the maritime schools. • To carry out literature studies on lifestyle intervention in the contexts • To describe and follow the prevalence of the target variables by age and gender over the years. (Prediabetes, T2DM, Blood pressure, BMI etc.) • To identify and offer the prediabetes remission program and to follow the effects. '''Data content''' Random samples of data extracted from the routine maritime health examinations in specified calendar periods. Variables • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer or non-officer on board • Blood pressure • Blood sugar HbA1c or Fasting Glucose, or Post prandial • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Participating in a life-style health promotion plan y/n Automatic recoding of data Excel-recoding formulas-Dec-1-2024 • Blood pressure: Normal, prehypertension, stage 1, 2 • Blood sugar: Normal, prediabetes, diabetes mellitus type 2, • BMI: normal, overweight, obese '''Blood-sugar-Measurements''' Either A1c mmol/mol or Fasting mg/dl (glucometer) or Postprandial sugar: mg/dl (glucometer) Validity of the method. (ref 2) Clinics without all the variables are invited to give the data they have. '''Use of the data for prevention''' To describe the prevalence of Prediabetes, T2DM, Hypertension, Prehypertension and BMI in the age- and gender groups and occupational groups. Possibly, to offer a prediabetes, prehypertension remission program. To propose (inter) national maritime health policies. To be used for the student’s theses. To publish in international health journals. '''Type of study •''' International, multicenter observational prevalence database, • Retrospective or Prospective longitudinal surveillance system using repeated routine maritime health prevalence examination data. • Includes an intervention component for individuals identified with prediabetes. '''Populations sources''' The database will include random samples of data extracted from routine health examinations conducted at: • Maritime clinics • Maritime schools Participants may include: • Seafarers • Fishers • Maritime students • Officers and non-officers '''Data collection''' Retrospectively, the study uses secondary data obtained from existing routine medical examinations but can also collect new research-specific measurements prospectively. Key variables include: • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer on board7 non officer • Blood pressure • Blood sugar HbA1c or Fasting Glucose • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Actually, included in a health promotion program '''Ethics and data governance Need for ethics review.''' In a retrospective desgn, formal ethics review is not required, if the study involves fully de-identified pre-existing medical records, with no direct participant contact and no collection of identifiable personal information. In accordance with institutional and national guidelines for minimal-risk research, this type of secondary data analysis is automatically exempt from ethics committee review. '''Ethical framework and legal compliance.''' The study will follow the Declaration of Helsinki, national research rules, the EU General Data Protection Regulation (GDPR) where it applies, and the rules of participating countries. Before enrolment or data transfer, the team must record who controls and processes the data and the legal reason for using health information. '''Consent and participant rights.''' Participants will receive clear information about the study’s length, health data, follow-up, record linking, international sharing, and approved future research. Consent will be obtained when required. An ethics committee must approve use of data without consent. Refusing or leaving will not affect healthcare, work, training, or education. No new data will be collected after withdrawal, but anonymous data or completed analyses may not be removable. Participants will be told this beforehand. '''Long-term follow-up and record linking.''' Each person will receive a study code, allowing records to be linked without putting names in the central database. Each clinic or school will keep the code key separately with restricted access. The central database will hold coded data, or anonymous data when linking is finished. Exact dates and unusual details will be grouped to reduce identification risk. Further contact or record links need legal and ethics approval. '''Where will data be stored.''' Data will be kept only on approved servers or encrypted research systems. Names and local code keys will remain at the original site in a separate, encrypted location. Personal computers, unencrypted devices, personal email, and unapproved cloud services are banned. Security includes encryption, backups, logs, malware protection, and an incident plan. Study documents must name the host institution and storage location. '''Who can access the data.''' Only named, authorised team members may access the minimum information needed for approved work. They must follow confidentiality rules, use individual accounts and secure sign-in, and lose access when their role ends. Collaborators may receive only necessary coded or anonymous data after approval. Employers, crewing companies, insurers, and other unauthorised parties will not receive personal-level data. Reports will hide small groups or identifying details. '''Data sharing, retention, and destruction.''' Institutions will sign agreements covering responsibilities, approved uses, security, breach reporting, further sharing, and data return or deletion. International transfers must follow the law and use extra safeguards when required. Before the study starts, the plan must state how long data will be kept (e.g. 5 years). Afterwards, files and backups will be securely deleted or made permanently anonymous. Code keys will be destroyed unless longer storage is approved. '''Risk management and oversight.''' The main risk is that private information could be exposed or linked to a person, causing work, social, or insurance harm. The team will collect only necessary data, separate names from research data, limit access, train staff, secure transfers and storage, keep logs, and respond quickly to suspected breaches. The coordinating researcher will record approvals, access permissions, agreements, and incidents. Any prevention or lifestyle programme tested as research needs ethics approval before it begins. '''Exclusion Criteria''' • Inability to provide informed consent • Expressed no-interest to participate '''The primary outcomes include''' • Prevalence of Prediabetes, and T2DM • Prevalence of Hypertension and Prehypertension • Prevalence of Overweight and Obesity • Trends by age and sex over time • Prevalence of participation in a prevention program Planned analyses • Estimate disease prevalence. • Compare health indicators across age, gender and working groups. • Monitor trends over multiple years. • Evaluate outcomes of lifestyle interventions. • Support health policy development, student research, and scientific publications. '''Statistical analysis plan''' ''1. Data preparation'' Check the dataset for missing values, impossible values, and duplicates. Recode the main variables into categories: BMI as normal, overweight, or obese; blood pressure as normal, prehypertension, stage 1, or stage 2; and blood sugar as normal, prediabetes, or type 2 diabetes. ''2. Descriptive statistics'' Summarize age using mean, standard deviation, median, and range. Summarize categorical variables such as gender, work type, smoking, medication use, BMI group, blood pressure group, and blood sugar group using numbers and percentages. These data is presented in the standard -29, 30-49. 50+ age- and gender groups. ''3. Prevalence estimates in the main age-groups.'' Calculate the prevalence of overweight, obesity, prediabetes, diabetes, prehypertension, and hypertension in the Report each prevalence as a percentage with a 95% confidence interval ''n the main age-groups -29, 30-49, 50+'' ''4. Group comparisons'' Compare prevalence by the 3 age group, gender, work type, and officer/non-officer status. Use chi-square tests for categorical comparisons and t-tests or ANOVA for continuous variables such as BMI or blood pressure. ''5. Trend analysis over time'' If data are available from several years, changes in prevalence over time using tables and line graphs. Test trends using chi-square trend tests or simple logistic regression with year as the predictor. ''6. Prediabetes remission program analysis'' Among participants offered the lifestyle intervention, compare blood sugar, BMI, and blood pressure before and after the program. Use paired t-tests for continuous measures and McNemar’s test for changes in categories, such as prediabetes returning to normal blood sugar. ''7. Missing data'' Report the amount of missing data for each variable. For the simple analysis, use available-case analysis and clearly state the number of participants included in each analysis. ''8. Statistical significance'' Use a significance level of 0.05. Present results with 95% confidence intervals rather than relying only on p-values.Principio del formulario '''Prevention plan''' '''Personal prevention''' reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association) '''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework) '''INFORMED CONSENT FORM''' International Maritime Health Database You are invited to participate in the International Maritime Health Database, which aims to improve the health of seafarers, fishers, and maritime students by monitoring conditions such as prediabetes, type 2 diabetes, hypertension, and obesity. If you agree to participate, information from your routine maritime medical examination (such as age, sex, height, weight, blood pressure, blood glucose, smoking status, medication, and occupation) will be included in a confidential research database. Your participation is voluntary. You may refuse to participate or withdraw your consent at any time without affecting your medical care, employment, or education. Your personal information will be kept confidential. Your data will be coded whenever possible and used only for research, education, and publication in a way that does not identify you. If your results indicate an increased risk of disease, you may be offered information about a voluntary health promotion or prevention programme. By signing below, you confirm that: • You have read and understood this information. • You have had the opportunity to ask questions. • You voluntarily agree to participate in this study. Participant Name: _______________________________ Signature: ______________________________________ Date: __________________________________________ Researcher/Health Professional: ___________________ Signature: ______________________________________ Date: __________________________________________ 7sw0x3qr9sf9ilhy8i3ue4pajgti8xo 2834996 2834859 2026-09-29T07:59:39Z JSINST 3110286 2834996 wikitext text/x-wiki ------------- '''Design of the project''' (Informed Consent Form below) This is an international, multicentre, observational longitudinal health database that uses routinely collected maritime health examination data to monitor the prevalence and trends of prediabetes, diabetes T-2, hypertension, prehypertension, overweight, obesity and established prevention among maritime workers and students. The intention is to register and distribute information on standardized prevention Inspired by the “Danish health database” (ref. 1)References 1. Health of the Danes https://www.danskernessundhed.dk/ '''Strengths of the design''' • Uses routinely collected data, making it relatively inexpensive. • Can accumulate large international datasets. • Reflects real-world clinical practice. • Supports long-term surveillance of maritime workers' health. • Allow for evaluation of preventive interventions within routine healthcare. Organisers• The John Snow Prediabetes Institute in collaboration with: • Maritime Medical Health Clinics, • The International Maritime Health Associtation (IMHA), • the Filippine Maritime Health Association '''Objectives''' • To establish an “International maritime health database” with samples of health examination data from the maritime clinics and the maritime schools. • To carry out literature studies on lifestyle intervention in the contexts • To describe and follow the prevalence of the target variables by age and gender over the years. (Prediabetes, T2DM, Blood pressure, BMI etc.) • To identify and offer the prediabetes remission program and to follow the effects. '''Data content''' Random samples of data extracted from the routine maritime health examinations in specified calendar periods. Variables • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer or non-officer on board • Blood pressure • Blood sugar HbA1c or Fasting Glucose, or Post prandial • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Participating in a life-style health promotion plan y/n Automatic recoding of data Excel-recoding formulas-Dec-1-2024 • Blood pressure: Normal, prehypertension, stage 1, 2 • Blood sugar: Normal, prediabetes, diabetes mellitus type 2, • BMI: normal, overweight, obese '''Blood-sugar-Measurements''' Either A1c mmol/mol or Fasting mg/dl (glucometer) or Postprandial sugar: mg/dl (glucometer) Validity of the method. (ref 2) Clinics without all the variables are invited to give the data they have. '''Use of the data for prevention''' To describe the prevalence of Prediabetes, T2DM, Hypertension, Prehypertension and BMI in the age- and gender groups and occupational groups. Possibly, to offer a prediabetes, prehypertension remission program. To propose (inter) national maritime health policies. To be used for the student’s theses. To publish in international health journals. '''Type of study •''' International, multicenter observational prevalence database, • Retrospective or Prospective longitudinal surveillance system using repeated routine maritime health prevalence examination data. • Includes an intervention component for individuals identified with prediabetes. '''Populations sources''' The database will include random samples of data extracted from routine health examinations conducted at: • Maritime clinics • Maritime schools Participants may include: • Seafarers • Fishers • Maritime students • Officers and non-officers '''Data collection''' Retrospectively, the study uses secondary data obtained from existing routine medical examinations but can also collect new research-specific measurements prospectively. Key variables include: • Age • Gender: m/f • Height: cm • Weight: kg or pounds • Work: fisher, seafarer, other (student) • Officer on board7 non officer • Blood pressure • Blood sugar HbA1c or Fasting Glucose • Smoking: y/n • Taking medicine: None, Metformin, Wegovy, other • Actually, included in a health promotion program '''Ethics and data governance Need for ethics review.''' In a retrospective desgn, formal ethics review is not required, if the study involves fully de-identified pre-existing medical records, with no direct participant contact and no collection of identifiable personal information. In accordance with institutional and national guidelines for minimal-risk research, this type of secondary data analysis is automatically exempt from ethics committee review. '''Ethical framework and legal compliance.''' The study will follow the Declaration of Helsinki, national research rules, the EU General Data Protection Regulation (GDPR) where it applies, and the rules of participating countries. Before enrolment or data transfer, the team must record who controls and processes the data and the legal reason for using health information. '''Consent and participant rights.''' Participants will receive clear information about the study’s length, health data, follow-up, record linking, international sharing, and approved future research. Consent will be obtained when required. An ethics committee must approve use of data without consent. Refusing or leaving will not affect healthcare, work, training, or education. No new data will be collected after withdrawal, but anonymous data or completed analyses may not be removable. Participants will be told this beforehand. '''Long-term follow-up and record linking.''' Each person will receive a study code, allowing records to be linked without putting names in the central database. Each clinic or school will keep the code key separately with restricted access. The central database will hold coded data, or anonymous data when linking is finished. Exact dates and unusual details will be grouped to reduce identification risk. Further contact or record links need legal and ethics approval. '''Where will data be stored.''' Data will be kept only on approved servers or encrypted research systems. Names and local code keys will remain at the original site in a separate, encrypted location. Personal computers, unencrypted devices, personal email, and unapproved cloud services are banned. Security includes encryption, backups, logs, malware protection, and an incident plan. Study documents must name the host institution and storage location. '''Who can access the data.''' Only named, authorised team members may access the minimum information needed for approved work. They must follow confidentiality rules, use individual accounts and secure sign-in, and lose access when their role ends. Collaborators may receive only necessary coded or anonymous data after approval. Employers, crewing companies, insurers, and other unauthorised parties will not receive personal-level data. Reports will hide small groups or identifying details. '''Data sharing, retention, and destruction.''' Institutions will sign agreements covering responsibilities, approved uses, security, breach reporting, further sharing, and data return or deletion. International transfers must follow the law and use extra safeguards when required. Before the study starts, the plan must state how long data will be kept (e.g. 5 years). Afterwards, files and backups will be securely deleted or made permanently anonymous. Code keys will be destroyed unless longer storage is approved. '''Risk management and oversight.''' The main risk is that private information could be exposed or linked to a person, causing work, social, or insurance harm. The team will collect only necessary data, separate names from research data, limit access, train staff, secure transfers and storage, keep logs, and respond quickly to suspected breaches. The coordinating researcher will record approvals, access permissions, agreements, and incidents. Any prevention or lifestyle programme tested as research needs ethics approval before it begins. '''Exclusion Criteria''' • Inability to provide informed consent • Expressed no-interest to participate '''The primary outcomes include''' • Prevalence of Prediabetes, and T2DM • Prevalence of Hypertension and Prehypertension • Prevalence of Overweight and Obesity • Trends by age and sex over time • Prevalence of participation in a prevention program Planned analyses • Estimate disease prevalence. • Compare health indicators across age, gender and working groups. • Monitor trends over multiple years. • Evaluate outcomes of lifestyle interventions. • Support health policy development, student research, and scientific publications. '''Statistical analysis plan''' ''1. Data preparation'' Check the dataset for missing values, impossible values, and duplicates. Recode the main variables into categories: BMI as normal, overweight, or obese; blood pressure as normal, prehypertension, stage 1, or stage 2; and blood sugar as normal, prediabetes, or type 2 diabetes. ''2. Descriptive statistics'' Summarize age using mean, standard deviation, median, and range. Summarize categorical variables such as gender, work type, smoking, medication use, BMI group, blood pressure group, and blood sugar group using numbers and percentages. These data is presented in the standard -29, 30-49. 50+ age- and gender groups. ''3. Prevalence estimates in the main age-groups.'' Calculate the prevalence of overweight, obesity, prediabetes, diabetes, prehypertension, and hypertension in the Report each prevalence as a percentage with a 95% confidence interval ''n the main age-groups -29, 30-49, 50+'' ''4. Group comparisons'' Compare prevalence by the 3 age group, gender, work type, and officer/non-officer status. Use chi-square tests for categorical comparisons and t-tests or ANOVA for continuous variables such as BMI or blood pressure. ''5. Trend analysis over time'' If data are available from several years, changes in prevalence over time using tables and line graphs. Test trends using chi-square trend tests or simple logistic regression with year as the predictor. ''6. Prediabetes remission program analysis'' Among participants offered the lifestyle intervention, compare blood sugar, BMI, and blood pressure before and after the program. Use paired t-tests for continuous measures and McNemar’s test for changes in categories, such as prediabetes returning to normal blood sugar. ''7. Missing data'' Report the amount of missing data for each variable. For the simple analysis, use available-case analysis and clearly state the number of participants included in each analysis. ''8. Statistical significance'' Use a significance level of 0.05. Present results with 95% confidence intervals rather than relying only on p-values.Principio del formulario '''Prevention plan''' '''Personal prevention''' reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (Contact your local Diabetes Association) '''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs (such as the Centers for Disease Control and Prevention framework) '''INFORMED CONSENT FORM''' International Maritime Health Database You are invited to participate in the International Maritime Health Database, which aims to improve the health of seafarers, fishers, and maritime students by monitoring conditions such as prediabetes, type 2 diabetes, hypertension, and obesity. If you agree to participate, information from your routine maritime medical examination (such as age, sex, height, weight, blood pressure, blood glucose, smoking status, medication, and occupation) will be included in a confidential research database. Your participation is voluntary. You may refuse to participate or withdraw your consent at any time without affecting your medical care, employment, or education. Your personal information will be kept confidential. Your data will be coded whenever possible and used only for research, education, and publication in a way that does not identify you. If your results indicate an increased risk of disease, you may be offered information about a voluntary health promotion or prevention programme. By signing below, you confirm that: • You have read and understood this information. • You have had the opportunity to ask questions. • You voluntarily agree to participate in this study. Participant Name: _______________________________ Signature: ______________________________________ Date: __________________________________________ Researcher/Health Professional: ___________________ Signature: ______________________________________ Date: __________________________________________ ge6svyejyvzopfpkpdim7dv5s6uz6sa 1 331279 2835014 2829575 2026-09-29T09:33:35Z U3253363 3106362 /* Suggestion to add a table */ new section 2835014 wikitext text/x-wiki == Focus questions == Promising progress. Remember to use bullet-points for the focus questions. See Tutorial 2. ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:09, 19 August 2026 (UTC) Really enjoyed this chapter! Curious since positive urgency and negative urgency are both part of the UPPS-P model, do you think one is more strongly linked to alcohol related consequences than the other, or does the research not really distinguish between them in that way? <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle were almost correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Heading structure --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Note that it isn't necessary to focus on alcohol (this is not part of the question), but alcohol can provide a useful case study/scenario/example |3= <!-- Overview--> # Very good <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section <!-- Description --> # An overaly detailed description of the problem/topic is planned or presented; most citations are best used in subsequent sections # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings |4= <!-- Key points--> <!-- Overall --> # Promising development <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad), but note that the question is not about alcohol per se <!-- Theory and research --> # Promising balance of theory and research # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Provide more detailed edit summaries <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is underway # Reduce emphasis on citations (these should be in previous sections); provide user-friendly, practical, take-home messages (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) presented and captioned, but not cited <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) # Place quiz each question in the most relevant section <!-- Tables --> # Promising use of table(s) |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses |8= <!-- Resources --> <!-- See also --> # See also ## Very good <!-- External links --> # External links ## Very good ## Move academic sources into the "References" sections and provide in-text citation ## Links to measurement instruments are probably not the best resources about this topic; link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent <!-- Description about self --> # Excellent description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:18, 29 August 2026 (UTC) == Suggestion to add a table == Hi there I read over your chapter and it is looking really good! In reading the UPPS-P introduction, I noticed the sentence detailing each dimension was quite long. To help readers' comprehension, I translated this sentence into a table. You may like to consider now condensing the sentence where you explain each dimension of the UPPS model and instead refer readers to Table 1. I hope that helps! All the best, Pepper. [[User:U3253363|U3253363]] ([[User talk:U3253363|discuss]] • [[Special:Contributions/U3253363|contribs]]) 09:33, 29 September 2026 (UTC) n00vtjj2okyntu0yu519pfa5yofrhrl 2 331305 2834992 2829300 2026-09-29T07:51:52Z U3239236 3106753 2834992 wikitext text/x-wiki == About me == I am studying [[Motivation and emotion]] and studying at the [http://Canberra.edu.au University of Canberra] Some of my hobbies include: * [[w:Reading|Reading]] * Gymming * Painting * Trying different cuisines == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Emotional intelligence and emotional wellbeing|Emotional Intelligence and Emotional Wellbeing]] == Social contributions == # [[Talk:Motivation and emotion/Book/2026/Romantic jealousy|Suggested reference]] # [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FImpulsivity_versus_sensation-seeking&diff=2825386&oldid=2823024 Asked a question about someone's topic] #[[Talk:Motivation and emotion/Book/2026/Value congruence and motivation|Left a comment on someone's book chapter]] #[https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FTechnology-based_pain_management&diff=2834991&oldid=2832299 Left feedback on a book chapter about technology-based pain management] dbdfknyqdgnv9tyq7nkgeouzyvlvng3 2834993 2834992 2026-09-29T07:55:09Z U3239236 3106753 added another social contribution 2834993 wikitext text/x-wiki == About me == I am studying [[Motivation and emotion]] and studying at the [http://Canberra.edu.au University of Canberra] Some of my hobbies include: * [[w:Reading|Reading]] * Gymming * Painting * Trying different cuisines == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Emotional intelligence and emotional wellbeing|Emotional Intelligence and Emotional Wellbeing]] == Social contributions == # [[Talk:Motivation and emotion/Book/2026/Romantic jealousy|Suggested reference]] # [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FImpulsivity_versus_sensation-seeking&diff=2825386&oldid=2823024 Asked a question about someone's topic] #[[Talk:Motivation and emotion/Book/2026/Value congruence and motivation|Left a comment on someone's book chapter]] #[https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FTechnology-based_pain_management&diff=2834991&oldid=2832299 Left feedback on a book chapter about technology-based pain management] #[https://uclearn.canberra.edu.au/courses/20143/discussion_topics/464129 Started a discussion about my book chapter] libk7yi96lh83i46bg1mtssm61y45e5 2 331388 2834963 2833829 2026-09-29T03:57:19Z P U3270518 3106535 /* Social contributions */ 2834963 wikitext text/x-wiki == About me == Hello Everyone, I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&rls=en&q=University+of+canberra&ie=UTF-8&oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit. My future goal is to become clinical psychologist and work with Australian Federal Police. My Linkedln profile: https://www.linkedin.com/in/palak-kathiriya-6b7048289/ Thank you for visiting my linkedln profile and this page! == Hobbies == * Dancing * Hiking * Travelling to different countries * Painting * Cycling * Nature exploration ** Birdwatching ** Gardening == Book Chapter I'm working on == I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] == Social contributions == All times are in Canberra local time (AEST/AEDT) # 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&diff=2826658&oldid=2826657 Added the template and the title for this page - For this chapter: Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)] # 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&diff=2826589&oldid=2826579 Fixed spelling error in one of the focus question - For this chapter: Perfectionism and procrastination - What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)] # 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - For this chapter: Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]] # 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)] #8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on discussion forum about how to use GenAI in the unit and what are the expectations if we use GenAI (UC Learn)] # 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&diff=2830323&oldid=2761410 Fixed spelling error in book chapter - For this chapter - Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)] # 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&diff=prev&oldid=2830325 Changed few sentences to make it grammatically better - For this chapter: Grit and academic achievement What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)] # 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830328&oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - For this chapter: Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)] # 10:54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830330&oldid=2830328 Added reference to support one claim in overview section - For this chapter: Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)] #12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)] #9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2830732&oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow - For this chapter: Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #9:27 pm, 3 September 2026: [[Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being|Provided suggestion for wikiversity book chapter page and clarified that the user page should be separate than the book chapter - For this chapter: Outdoor play and children's emotional well-being - How does outdoor play influence children's emotional well-being? Overview - (Book Chapter, 2025)]] #8:28 am, 4 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 Commented on discussion forum about what motivates me and what helps me to keep going if I am not feeling motivated (UC Learn)] #8:30 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2831894&oldid=2830732 Changed few spelling errors, fixed sentence structure and punctuation, shortened some repetitive wording while keeping original ideas - For this chapter- Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #8:50 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2831919&oldid=2830330 Changed few sentences to make it grammatically correct and to maintain sentence flow - For this chapter - Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake? - (Book chapter, 2025)] #8:34 pm, 8 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832227&oldid=2804069 Changed few sentences to make it grammatically correct and added citations to support few claims - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use?- (Book chapter, 2025)] #8:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832228&oldid=2832227 Added citations to support claims in overview section - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use ? (Book chapter, 2025)] #8:59 pm, 8 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455064 Commented on UC Learn discussion forum to connect with peers and also shared my Linkedln profile] #9:19 pm, 8 September 2026: [[Talk:Motivation and emotion/Book/2026/Warm-glow giving|Provided suggestion about overview section on talk page - For this chapter - Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour?- (Book chapter, 2026)]] #9:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FSubcortical_structures_and_motivational_drive#See_also_and_external_links_section Commented on talk page to improve see also, external links, and conclusion section - For this chapter - Subcortical structures and motivational drive: How do subcortical brain regions generate basic motivational impulses and energy?- (Book chapter, 2026)] #3:07 pm , 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FEnvironmental_volunteering_motivation&diff=2832334&oldid=2814930 Corrected few grammatical errors, added citation to few important claims, provided an introductory paragraph in environmental volunteering section, explained few points in detail, added volume number in one of the references - For this chapter - Environmental volunteering motivation: What motivates environmental volunteering (Book chapter, 2022)] #3:19 pm, 9 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Commented on UC Learn about a book chapter question related to value congruence and motivation] #9:35 pm, 9 September 2026: [[Talk:Motivation and emotion/Book/2026/Alcohol use for emotion regulation|Provided suggestion about book chapter that chapter should include both positive and negative dysregulation - For this chapter - Alcohol use for emotion regulation : why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)]] #9:45 pm, 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAlcohol_use_for_emotion_regulation&diff=2832417&oldid=2830147 Fixed title for this chapter - For this chapter - Alcohol use for emotion regulation: why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)] #10:00 am, 11 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461216 Contributed in discussion forum and answered question about word count (UC Learn)] #10:40 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FE-cigarette_use_motivation&diff=2832747&oldid=2754142 Used bullet points to organise see also and external link section and included source in parentheses - For this chapter - E-cigarettes use motivation: What motivates starting and continuing vaping of nicotine e-cigarettes? - (Book chapter, 2024)] #11:01 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_and_ego_depletion&diff=2832750&oldid=2741278 Corrected grammar and spelling errors - For this chapter - Sleep and ego depletion: How does sleep affect the capacity for self-control and willpower? - (Book chapter, 2024)] #11:14 pm, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2020%2FChild_killer_motivation&diff=2832817&oldid=2721628 Used bullet points to organise see also section and provided few external sources - For this chapter - Child killer motivation: What motivates a child to kill? (Book chapter, 2020)] #9:02 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FBurnout&diff=2832924&oldid=2666399 Fixed overcapitalisation in focus questions, moved one reference into references section and fixed see also section - For this chapter - Burnout: What is burnout and how can be it managed and prevented - (Book chapter, 2022)] #9:37 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FTo-do_lists&diff=2832933&oldid=2655448 Corrected grammar and spelling errors - For this chapter -To-do lists: Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - (Book chapter, 2022)] #8:17 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAbusive_relationships_and_emotional_dependency&diff=2833144&oldid=2723675 Rewrote sentence to make it more clear to understand - For this chapter - Abusive relationships and emotional dependency: What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?(Book chapter, 2024)] #8:34 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FDopamine_fasting&diff=2833148&oldid=2716222 Rewrote few sentences to make it grammatically correct- For this chapter -Dopamine fasting: What is dopamine fasting and how does it affect motivation and emotion?- (Book chapter, 2024)] #8:41 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2018%2FBody_image_and_emotional_well-being&diff=2833150&oldid=2596066 Improved see also section - For this chapter - Body image and emotional well-being: How does body image influence one's emotional well being?- (Book chapter, 2018)] #8:47 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FClimate_change_helplessness&diff=2833152&oldid=2596067 Corrected few spellings and external links section - For this chapter -Climate change helplessness: How does learned helplessness impact motivation to engage in behaviours to limit climate change?(Book chapter, 2021)] #9:05 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FDental_fear&diff=2833154&oldid=2649741 Corrected grammatical errors - For this chapter -Dental fear: What causes dental fear, what are the consequences, and how can it be managed? - (Book chapter, 2021)] #7:01 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 Corrected 2 spelling mistake - For this chapter- Psychedelic treatment of eating disorders:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 How might psychedelic-assisted therapy influence psychological mechanisms involved in eating disorders? - (Book chapter, 2026)] #7:14 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWisdom%2C_motivation%2C_and_emotion&diff=2833301&oldid=2831617 Corrected spelling error - For this chapter - Wisdom, motivation, and emotion: How do motivational and emotional processes contribute to wisdom? - (Book chapter, 2026)] #11:05 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FGender_and_achievement_motivation&diff=2833381&oldid=2832014 Corrected few grammatical errors - For this Chapter - Gender and achievement motivation: How does gender shape where, how, and under what conditions achievement motivation is expressed? - (Book chapter, 2026)] #11:23 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FVolunteer_wellbeing&diff=2833387&oldid=2829983 Corrected grammar error - For this chapter -Volunteer wellbeing: How does volunteering affect volunteers' subjective wellbeing? - (Book chapter, 2026)] #7: 38 pm, 16 September 2026: [[Talk:Motivation and emotion/Book/2026/Creative inspiration and effort|Commented on talk page and provided suggestion about theory that can be included in the chapter - For this chapter - Creative inspiration and effort: How do inspiration and effort interact during the creative process?]] #3:35 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 Corrected 2 spellings - For this chapter - Alcohol, dopamine, motivation, and emotion:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 What role does dopamine play in motivational and emotional responses to alcohol consumption? - (Book chapter, 2024)] #3:48 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FVasopressin_and_motivation&diff=2833579&oldid=2729665 Corrected spelling error - For this chapter - Vasopressin and motivation: How does vasopressin influence motivation? - (Book chapter, 2024)] #3:53 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_onset_optimisation&diff=2833580&oldid=2687478 Corrected 2 spellings - For this chapter - Sleep onset optimisation : What techniques assist in falling asleep promptly? - (Book chapter, 2024)] #2:12 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FProcrastination_and_temporal_discounting&diff=2833689&oldid=2761864 Corrected spelling errors - For this chapter - Procastination and temporal discounting: How does valuing immediate rewards over future outcomes contribute to procrastination? - (Book chapter, 2025)] #2:20 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMaladaptive_defence_mechanisms_and_distress&diff=2833691&oldid=2761577 Corrected spelling errors - For this chapter- Maladaptive defence mechanisms and distress: In what ways do maladaptive defences shape emotional regulation and behavioural responses when under pressure? - (Book chapter, 2025)] #11:02 am, 19 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FSublimation&diff=2833828&oldid=2582187 Corrected 3 spelling errors - For this chapter -Sublimation: How can Sublimation be fostered? (Book chapter, 2023)] #1: 50 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FInner_voice_in_anxiety&diff=2834960&oldid=2761762 Corrected spelling error - For this chapter- Inner voice in anxiety: What role does the inner voice play in maintaining or alleviating anxiety? - (Book chapter, 2025)] qmwyw815i1hdyx0cvsb4f1y34w94e33 2834970 2834963 2026-09-29T07:02:08Z P U3270518 3106535 /* Social contributions */ 2834970 wikitext text/x-wiki == About me == Hello Everyone, I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&rls=en&q=University+of+canberra&ie=UTF-8&oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit. My future goal is to become clinical psychologist and work with Australian Federal Police. My Linkedln profile: https://www.linkedin.com/in/palak-kathiriya-6b7048289/ Thank you for visiting my linkedln profile and this page! == Hobbies == * Dancing * Hiking * Travelling to different countries * Painting * Cycling * Nature exploration ** Birdwatching ** Gardening == Book Chapter I'm working on == I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] == Social contributions == All times are in Canberra local time (AEST/AEDT) # 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&diff=2826658&oldid=2826657 Added the template and the title for this page - For this chapter: Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)] # 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&diff=2826589&oldid=2826579 Fixed spelling error in one of the focus question - For this chapter: Perfectionism and procrastination - What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)] # 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - For this chapter: Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]] # 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)] #8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on discussion forum about how to use GenAI in the unit and what are the expectations if we use GenAI (UC Learn)] # 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&diff=2830323&oldid=2761410 Fixed spelling error in book chapter - For this chapter - Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)] # 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&diff=prev&oldid=2830325 Changed few sentences to make it grammatically better - For this chapter: Grit and academic achievement What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)] # 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830328&oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - For this chapter: Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)] # 10:54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830330&oldid=2830328 Added reference to support one claim in overview section - For this chapter: Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)] #12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)] #9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2830732&oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow - For this chapter: Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #9:27 pm, 3 September 2026: [[Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being|Provided suggestion for wikiversity book chapter page and clarified that the user page should be separate than the book chapter - For this chapter: Outdoor play and children's emotional well-being - How does outdoor play influence children's emotional well-being? Overview - (Book Chapter, 2025)]] #8:28 am, 4 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 Commented on discussion forum about what motivates me and what helps me to keep going if I am not feeling motivated (UC Learn)] #8:30 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2831894&oldid=2830732 Changed few spelling errors, fixed sentence structure and punctuation, shortened some repetitive wording while keeping original ideas - For this chapter- Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #8:50 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2831919&oldid=2830330 Changed few sentences to make it grammatically correct and to maintain sentence flow - For this chapter - Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake? - (Book chapter, 2025)] #8:34 pm, 8 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832227&oldid=2804069 Changed few sentences to make it grammatically correct and added citations to support few claims - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use?- (Book chapter, 2025)] #8:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832228&oldid=2832227 Added citations to support claims in overview section - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use ? (Book chapter, 2025)] #8:59 pm, 8 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455064 Commented on UC Learn discussion forum to connect with peers and also shared my Linkedln profile] #9:19 pm, 8 September 2026: [[Talk:Motivation and emotion/Book/2026/Warm-glow giving|Provided suggestion about overview section on talk page - For this chapter - Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour?- (Book chapter, 2026)]] #9:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FSubcortical_structures_and_motivational_drive#See_also_and_external_links_section Commented on talk page to improve see also, external links, and conclusion section - For this chapter - Subcortical structures and motivational drive: How do subcortical brain regions generate basic motivational impulses and energy?- (Book chapter, 2026)] #3:07 pm , 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FEnvironmental_volunteering_motivation&diff=2832334&oldid=2814930 Corrected few grammatical errors, added citation to few important claims, provided an introductory paragraph in environmental volunteering section, explained few points in detail, added volume number in one of the references - For this chapter - Environmental volunteering motivation: What motivates environmental volunteering (Book chapter, 2022)] #3:19 pm, 9 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Commented on UC Learn about a book chapter question related to value congruence and motivation] #9:35 pm, 9 September 2026: [[Talk:Motivation and emotion/Book/2026/Alcohol use for emotion regulation|Provided suggestion about book chapter that chapter should include both positive and negative dysregulation - For this chapter - Alcohol use for emotion regulation : why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)]] #9:45 pm, 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAlcohol_use_for_emotion_regulation&diff=2832417&oldid=2830147 Fixed title for this chapter - For this chapter - Alcohol use for emotion regulation: why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)] #10:00 am, 11 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461216 Contributed in discussion forum and answered question about word count (UC Learn)] #10:40 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FE-cigarette_use_motivation&diff=2832747&oldid=2754142 Used bullet points to organise see also and external link section and included source in parentheses - For this chapter - E-cigarettes use motivation: What motivates starting and continuing vaping of nicotine e-cigarettes? - (Book chapter, 2024)] #11:01 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_and_ego_depletion&diff=2832750&oldid=2741278 Corrected grammar and spelling errors - For this chapter - Sleep and ego depletion: How does sleep affect the capacity for self-control and willpower? - (Book chapter, 2024)] #11:14 pm, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2020%2FChild_killer_motivation&diff=2832817&oldid=2721628 Used bullet points to organise see also section and provided few external sources - For this chapter - Child killer motivation: What motivates a child to kill? (Book chapter, 2020)] #9:02 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FBurnout&diff=2832924&oldid=2666399 Fixed overcapitalisation in focus questions, moved one reference into references section and fixed see also section - For this chapter - Burnout: What is burnout and how can be it managed and prevented - (Book chapter, 2022)] #9:37 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FTo-do_lists&diff=2832933&oldid=2655448 Corrected grammar and spelling errors - For this chapter -To-do lists: Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - (Book chapter, 2022)] #8:17 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAbusive_relationships_and_emotional_dependency&diff=2833144&oldid=2723675 Rewrote sentence to make it more clear to understand - For this chapter - Abusive relationships and emotional dependency: What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?(Book chapter, 2024)] #8:34 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FDopamine_fasting&diff=2833148&oldid=2716222 Rewrote few sentences to make it grammatically correct- For this chapter -Dopamine fasting: What is dopamine fasting and how does it affect motivation and emotion?- (Book chapter, 2024)] #8:41 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2018%2FBody_image_and_emotional_well-being&diff=2833150&oldid=2596066 Improved see also section - For this chapter - Body image and emotional well-being: How does body image influence one's emotional well being?- (Book chapter, 2018)] #8:47 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FClimate_change_helplessness&diff=2833152&oldid=2596067 Corrected few spellings and external links section - For this chapter -Climate change helplessness: How does learned helplessness impact motivation to engage in behaviours to limit climate change?(Book chapter, 2021)] #9:05 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FDental_fear&diff=2833154&oldid=2649741 Corrected grammatical errors - For this chapter -Dental fear: What causes dental fear, what are the consequences, and how can it be managed? - (Book chapter, 2021)] #7:01 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 Corrected 2 spelling mistake - For this chapter- Psychedelic treatment of eating disorders:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 How might psychedelic-assisted therapy influence psychological mechanisms involved in eating disorders? - (Book chapter, 2026)] #7:14 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWisdom%2C_motivation%2C_and_emotion&diff=2833301&oldid=2831617 Corrected spelling error - For this chapter - Wisdom, motivation, and emotion: How do motivational and emotional processes contribute to wisdom? - (Book chapter, 2026)] #11:05 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FGender_and_achievement_motivation&diff=2833381&oldid=2832014 Corrected few grammatical errors - For this Chapter - Gender and achievement motivation: How does gender shape where, how, and under what conditions achievement motivation is expressed? - (Book chapter, 2026)] #11:23 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FVolunteer_wellbeing&diff=2833387&oldid=2829983 Corrected grammar error - For this chapter -Volunteer wellbeing: How does volunteering affect volunteers' subjective wellbeing? - (Book chapter, 2026)] #7: 38 pm, 16 September 2026: [[Talk:Motivation and emotion/Book/2026/Creative inspiration and effort|Commented on talk page and provided suggestion about theory that can be included in the chapter - For this chapter - Creative inspiration and effort: How do inspiration and effort interact during the creative process?]] #3:35 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 Corrected 2 spellings - For this chapter - Alcohol, dopamine, motivation, and emotion:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 What role does dopamine play in motivational and emotional responses to alcohol consumption? - (Book chapter, 2024)] #3:48 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FVasopressin_and_motivation&diff=2833579&oldid=2729665 Corrected spelling error - For this chapter - Vasopressin and motivation: How does vasopressin influence motivation? - (Book chapter, 2024)] #3:53 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_onset_optimisation&diff=2833580&oldid=2687478 Corrected 2 spellings - For this chapter - Sleep onset optimisation : What techniques assist in falling asleep promptly? - (Book chapter, 2024)] #2:12 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FProcrastination_and_temporal_discounting&diff=2833689&oldid=2761864 Corrected spelling errors - For this chapter - Procastination and temporal discounting: How does valuing immediate rewards over future outcomes contribute to procrastination? - (Book chapter, 2025)] #2:20 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMaladaptive_defence_mechanisms_and_distress&diff=2833691&oldid=2761577 Corrected spelling errors - For this chapter- Maladaptive defence mechanisms and distress: In what ways do maladaptive defences shape emotional regulation and behavioural responses when under pressure? - (Book chapter, 2025)] #11:02 am, 19 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FSublimation&diff=2833828&oldid=2582187 Corrected 3 spelling errors - For this chapter -Sublimation: How can Sublimation be fostered? (Book chapter, 2023)] #1:50 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FInner_voice_in_anxiety&diff=2834960&oldid=2761762 Corrected spelling error - For this chapter- Inner voice in anxiety: What role does the inner voice play in maintaining or alleviating anxiety? - (Book chapter, 2025)] #4:52 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FMultiple_selves_theory%2C_motivation%2C_and_emotion&diff=2834969&oldid=2755077 Corrected grammar - For this chapter - Multiple selves, motivation, and emotion: How does the concept of multiple selves impact our motivational and emotional experiences? - (Book chapter, 2023)] s7fh6y3jn3kdn7ewoz0oq1jl3r93mb1 2834972 2834970 2026-09-29T07:12:47Z P U3270518 3106535 /* Social contributions */ 2834972 wikitext text/x-wiki == About me == Hello Everyone, I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&rls=en&q=University+of+canberra&ie=UTF-8&oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit. My future goal is to become clinical psychologist and work with Australian Federal Police. My Linkedln profile: https://www.linkedin.com/in/palak-kathiriya-6b7048289/ Thank you for visiting my linkedln profile and this page! == Hobbies == * Dancing * Hiking * Travelling to different countries * Painting * Cycling * Nature exploration ** Birdwatching ** Gardening == Book Chapter I'm working on == I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] == Social contributions == All times are in Canberra local time (AEST/AEDT) # 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&diff=2826658&oldid=2826657 Added the template and the title for this page - For this chapter: Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)] # 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&diff=2826589&oldid=2826579 Fixed spelling error in one of the focus question - For this chapter: Perfectionism and procrastination - What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)] # 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - For this chapter: Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]] # 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)] #8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on discussion forum about how to use GenAI in the unit and what are the expectations if we use GenAI (UC Learn)] # 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&diff=2830323&oldid=2761410 Fixed spelling error in book chapter - For this chapter - Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)] # 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&diff=prev&oldid=2830325 Changed few sentences to make it grammatically better - For this chapter: Grit and academic achievement What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)] # 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830328&oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - For this chapter: Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)] # 10:54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830330&oldid=2830328 Added reference to support one claim in overview section - For this chapter: Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)] #12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)] #9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2830732&oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow - For this chapter: Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #9:27 pm, 3 September 2026: [[Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being|Provided suggestion for wikiversity book chapter page and clarified that the user page should be separate than the book chapter - For this chapter: Outdoor play and children's emotional well-being - How does outdoor play influence children's emotional well-being? Overview - (Book Chapter, 2025)]] #8:28 am, 4 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 Commented on discussion forum about what motivates me and what helps me to keep going if I am not feeling motivated (UC Learn)] #8:30 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2831894&oldid=2830732 Changed few spelling errors, fixed sentence structure and punctuation, shortened some repetitive wording while keeping original ideas - For this chapter- Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #8:50 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2831919&oldid=2830330 Changed few sentences to make it grammatically correct and to maintain sentence flow - For this chapter - Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake? - (Book chapter, 2025)] #8:34 pm, 8 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832227&oldid=2804069 Changed few sentences to make it grammatically correct and added citations to support few claims - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use?- (Book chapter, 2025)] #8:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832228&oldid=2832227 Added citations to support claims in overview section - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use ? (Book chapter, 2025)] #8:59 pm, 8 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455064 Commented on UC Learn discussion forum to connect with peers and also shared my Linkedln profile] #9:19 pm, 8 September 2026: [[Talk:Motivation and emotion/Book/2026/Warm-glow giving|Provided suggestion about overview section on talk page - For this chapter - Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour?- (Book chapter, 2026)]] #9:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FSubcortical_structures_and_motivational_drive#See_also_and_external_links_section Commented on talk page to improve see also, external links, and conclusion section - For this chapter - Subcortical structures and motivational drive: How do subcortical brain regions generate basic motivational impulses and energy?- (Book chapter, 2026)] #3:07 pm , 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FEnvironmental_volunteering_motivation&diff=2832334&oldid=2814930 Corrected few grammatical errors, added citation to few important claims, provided an introductory paragraph in environmental volunteering section, explained few points in detail, added volume number in one of the references - For this chapter - Environmental volunteering motivation: What motivates environmental volunteering (Book chapter, 2022)] #3:19 pm, 9 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Commented on UC Learn about a book chapter question related to value congruence and motivation] #9:35 pm, 9 September 2026: [[Talk:Motivation and emotion/Book/2026/Alcohol use for emotion regulation|Provided suggestion about book chapter that chapter should include both positive and negative dysregulation - For this chapter - Alcohol use for emotion regulation : why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)]] #9:45 pm, 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAlcohol_use_for_emotion_regulation&diff=2832417&oldid=2830147 Fixed title for this chapter - For this chapter - Alcohol use for emotion regulation: why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)] #10:00 am, 11 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461216 Contributed in discussion forum and answered question about word count (UC Learn)] #10:40 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FE-cigarette_use_motivation&diff=2832747&oldid=2754142 Used bullet points to organise see also and external link section and included source in parentheses - For this chapter - E-cigarettes use motivation: What motivates starting and continuing vaping of nicotine e-cigarettes? - (Book chapter, 2024)] #11:01 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_and_ego_depletion&diff=2832750&oldid=2741278 Corrected grammar and spelling errors - For this chapter - Sleep and ego depletion: How does sleep affect the capacity for self-control and willpower? - (Book chapter, 2024)] #11:14 pm, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2020%2FChild_killer_motivation&diff=2832817&oldid=2721628 Used bullet points to organise see also section and provided few external sources - For this chapter - Child killer motivation: What motivates a child to kill? (Book chapter, 2020)] #9:02 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FBurnout&diff=2832924&oldid=2666399 Fixed overcapitalisation in focus questions, moved one reference into references section and fixed see also section - For this chapter - Burnout: What is burnout and how can be it managed and prevented - (Book chapter, 2022)] #9:37 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FTo-do_lists&diff=2832933&oldid=2655448 Corrected grammar and spelling errors - For this chapter -To-do lists: Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - (Book chapter, 2022)] #8:17 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAbusive_relationships_and_emotional_dependency&diff=2833144&oldid=2723675 Rewrote sentence to make it more clear to understand - For this chapter - Abusive relationships and emotional dependency: What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?(Book chapter, 2024)] #8:34 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FDopamine_fasting&diff=2833148&oldid=2716222 Rewrote few sentences to make it grammatically correct- For this chapter -Dopamine fasting: What is dopamine fasting and how does it affect motivation and emotion?- (Book chapter, 2024)] #8:41 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2018%2FBody_image_and_emotional_well-being&diff=2833150&oldid=2596066 Improved see also section - For this chapter - Body image and emotional well-being: How does body image influence one's emotional well being?- (Book chapter, 2018)] #8:47 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FClimate_change_helplessness&diff=2833152&oldid=2596067 Corrected few spellings and external links section - For this chapter -Climate change helplessness: How does learned helplessness impact motivation to engage in behaviours to limit climate change?(Book chapter, 2021)] #9:05 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FDental_fear&diff=2833154&oldid=2649741 Corrected grammatical errors - For this chapter -Dental fear: What causes dental fear, what are the consequences, and how can it be managed? - (Book chapter, 2021)] #7:01 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 Corrected 2 spelling mistake - For this chapter- Psychedelic treatment of eating disorders:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 How might psychedelic-assisted therapy influence psychological mechanisms involved in eating disorders? - (Book chapter, 2026)] #7:14 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWisdom%2C_motivation%2C_and_emotion&diff=2833301&oldid=2831617 Corrected spelling error - For this chapter - Wisdom, motivation, and emotion: How do motivational and emotional processes contribute to wisdom? - (Book chapter, 2026)] #11:05 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FGender_and_achievement_motivation&diff=2833381&oldid=2832014 Corrected few grammatical errors - For this Chapter - Gender and achievement motivation: How does gender shape where, how, and under what conditions achievement motivation is expressed? - (Book chapter, 2026)] #11:23 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FVolunteer_wellbeing&diff=2833387&oldid=2829983 Corrected grammar error - For this chapter -Volunteer wellbeing: How does volunteering affect volunteers' subjective wellbeing? - (Book chapter, 2026)] #7: 38 pm, 16 September 2026: [[Talk:Motivation and emotion/Book/2026/Creative inspiration and effort|Commented on talk page and provided suggestion about theory that can be included in the chapter - For this chapter - Creative inspiration and effort: How do inspiration and effort interact during the creative process?]] #3:35 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 Corrected 2 spellings - For this chapter - Alcohol, dopamine, motivation, and emotion:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 What role does dopamine play in motivational and emotional responses to alcohol consumption? - (Book chapter, 2024)] #3:48 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FVasopressin_and_motivation&diff=2833579&oldid=2729665 Corrected spelling error - For this chapter - Vasopressin and motivation: How does vasopressin influence motivation? - (Book chapter, 2024)] #3:53 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_onset_optimisation&diff=2833580&oldid=2687478 Corrected 2 spellings - For this chapter - Sleep onset optimisation : What techniques assist in falling asleep promptly? - (Book chapter, 2024)] #2:12 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FProcrastination_and_temporal_discounting&diff=2833689&oldid=2761864 Corrected spelling errors - For this chapter - Procastination and temporal discounting: How does valuing immediate rewards over future outcomes contribute to procrastination? - (Book chapter, 2025)] #2:20 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMaladaptive_defence_mechanisms_and_distress&diff=2833691&oldid=2761577 Corrected spelling errors - For this chapter- Maladaptive defence mechanisms and distress: In what ways do maladaptive defences shape emotional regulation and behavioural responses when under pressure? - (Book chapter, 2025)] #11:02 am, 19 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FSublimation&diff=2833828&oldid=2582187 Corrected 3 spelling errors - For this chapter -Sublimation: How can Sublimation be fostered? (Book chapter, 2023)] #1:50 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FInner_voice_in_anxiety&diff=2834960&oldid=2761762 Corrected spelling error - For this chapter- Inner voice in anxiety: What role does the inner voice play in maintaining or alleviating anxiety? - (Book chapter, 2025)] #4:52 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FMultiple_selves_theory%2C_motivation%2C_and_emotion&diff=2834969&oldid=2755077 Corrected grammar - For this chapter - Multiple selves, motivation, and emotion: How does the concept of multiple selves impact our motivational and emotional experiences? - (Book chapter, 2023)] #5:07 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FEmotional_eating&diff=2834971&oldid=2573916 Corrected spelling error - For this chapter - Emotional eating: What is emotional eating and what effect does it have on emotion and health?- (Book chapter, 2021)] 388wkyb9v1r1ruwfdvjy7009xvac521 2834974 2834972 2026-09-29T07:24:01Z P U3270518 3106535 /* Social contributions */ 2834974 wikitext text/x-wiki == About me == Hello Everyone, I am currently a 3rd year psychology student at [https://www.google.com/search?client=safari&rls=en&q=University+of+canberra&ie=UTF-8&oe=UTF-8 University of Canberra] This semester I am studying [[Motivation and emotion|Motivation and Emotion]] unit. My future goal is to become clinical psychologist and work with Australian Federal Police. My Linkedln profile: https://www.linkedin.com/in/palak-kathiriya-6b7048289/ Thank you for visiting my linkedln profile and this page! == Hobbies == * Dancing * Hiking * Travelling to different countries * Painting * Cycling * Nature exploration ** Birdwatching ** Gardening == Book Chapter I'm working on == I am working on a really interesting topic for my book chapter. My chapter title is: Positive emotion dysregulation: What is positive emotion dysregulation and how does it affect psychological functioning? Link for my book chapter: [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] == Social contributions == All times are in Canberra local time (AEST/AEDT) # 12:58 pm, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWarm-glow_giving&diff=2826658&oldid=2826657 Added the template and the title for this page - For this chapter: Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour? '''('''Book Chapter, 2026)] # 11:59 am, 26 August 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPerfectionism_and_procrastination&diff=2826589&oldid=2826579 Fixed spelling error in one of the focus question - For this chapter: Perfectionism and procrastination - What is the role of perfectionism in procrastination and what can be done about it? (Book Chapter, 2026)] # 12:28 pm, 26 August 2026: [[Talk:Motivation and emotion/Book/2026/Motivations for using sex work services#Heading casing|Made a suggestion about overview section - For this chapter: Motivations for using sex work services: What motivates use of sex work services? (Book Chapter, 2026)]] # 2:03 pm, 26 August 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558 Provided a couple of sources for the development of breathing exercises and relaxation book chapter (Book Chapter, 2026) (UC Learn)] #8:10 am, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/458145 Commented on discussion forum about how to use GenAI in the unit and what are the expectations if we use GenAI (UC Learn)] # 10:14 am , 2 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSelf-control_and_ego_depletion_recovery&diff=2830323&oldid=2761410 Fixed spelling error in book chapter - For this chapter - Self-control and ego depletion recovery: How do people restore self-control resources after depletion and what factors influence recovery? (Book Chapter, 2025)] # 10:22 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Grit_and_academic_achievement&diff=prev&oldid=2830325 Changed few sentences to make it grammatically better - For this chapter: Grit and academic achievement What role does grit play in academic achievement and can it be fostered in future students? (Book Chapter, 2025)] # 10:48 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830328&oldid=2823381 Rewrote one paragraph in overview section to make it easier to understand - For this chapter: Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake?(Book Chapter, 2025)] # 10:54 am, 2 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2830330&oldid=2830328 Added reference to support one claim in overview section - For this chapter: Cancer screening and emotion:How do emotions such as fear, anxiety, and relief influence cancer screening uptake? (Book Chapter, 2025)] #12:56 pm, 2 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Commented on discussion forum about what interests me more in motivation and emotion unit and what I would like to learn (UC Learn)] #9:17 pm, 3 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2830732&oldid=2830326 Corrected grammar and sentence structure, fixed spelling, removed repetition, and improved wording for clarity and flow - For this chapter: Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #9:27 pm, 3 September 2026: [[Talk:Motivation and emotion/Book/2026/Outdoor play and children's emotional well-being|Provided suggestion for wikiversity book chapter page and clarified that the user page should be separate than the book chapter - For this chapter: Outdoor play and children's emotional well-being - How does outdoor play influence children's emotional well-being? Overview - (Book Chapter, 2025)]] #8:28 am, 4 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345 Commented on discussion forum about what motivates me and what helps me to keep going if I am not feeling motivated (UC Learn)] #8:30 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FGrit_and_academic_achievement&diff=2831894&oldid=2830732 Changed few spelling errors, fixed sentence structure and punctuation, shortened some repetitive wording while keeping original ideas - For this chapter- Grit and academic achievement - What role does grit play in academic achievement and can it be fostered in future students? - (Book Chapter, 2025)] #8:50 am, 7 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCancer_screening_and_emotion&diff=2831919&oldid=2830330 Changed few sentences to make it grammatically correct and to maintain sentence flow - For this chapter - Cancer screening and emotion: How do emotions such as fear, anxiety, and relief influence cancer screening uptake? - (Book chapter, 2025)] #8:34 pm, 8 September 2026:[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832227&oldid=2804069 Changed few sentences to make it grammatically correct and added citations to support few claims - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use?- (Book chapter, 2025)] #8:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2832228&oldid=2832227 Added citations to support claims in overview section - For this chapter - Boredom and substance use: What role does boredom play in motivating substance use ? (Book chapter, 2025)] #8:59 pm, 8 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455064 Commented on UC Learn discussion forum to connect with peers and also shared my Linkedln profile] #9:19 pm, 8 September 2026: [[Talk:Motivation and emotion/Book/2026/Warm-glow giving|Provided suggestion about overview section on talk page - For this chapter - Warm-glow giving - Why does giving feel good and how does this influence prosocial behaviour?- (Book chapter, 2026)]] #9:43 pm, 8 September 2026: [https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2026%2FSubcortical_structures_and_motivational_drive#See_also_and_external_links_section Commented on talk page to improve see also, external links, and conclusion section - For this chapter - Subcortical structures and motivational drive: How do subcortical brain regions generate basic motivational impulses and energy?- (Book chapter, 2026)] #3:07 pm , 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FEnvironmental_volunteering_motivation&diff=2832334&oldid=2814930 Corrected few grammatical errors, added citation to few important claims, provided an introductory paragraph in environmental volunteering section, explained few points in detail, added volume number in one of the references - For this chapter - Environmental volunteering motivation: What motivates environmental volunteering (Book chapter, 2022)] #3:19 pm, 9 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457190 Commented on UC Learn about a book chapter question related to value congruence and motivation] #9:35 pm, 9 September 2026: [[Talk:Motivation and emotion/Book/2026/Alcohol use for emotion regulation|Provided suggestion about book chapter that chapter should include both positive and negative dysregulation - For this chapter - Alcohol use for emotion regulation : why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)]] #9:45 pm, 9 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAlcohol_use_for_emotion_regulation&diff=2832417&oldid=2830147 Fixed title for this chapter - For this chapter - Alcohol use for emotion regulation: why and how people use alcohol to regulate their emotions? - (Book chapter, 2026)] #10:00 am, 11 September 2026: [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/461216 Contributed in discussion forum and answered question about word count (UC Learn)] #10:40 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FE-cigarette_use_motivation&diff=2832747&oldid=2754142 Used bullet points to organise see also and external link section and included source in parentheses - For this chapter - E-cigarettes use motivation: What motivates starting and continuing vaping of nicotine e-cigarettes? - (Book chapter, 2024)] #11:01 am, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_and_ego_depletion&diff=2832750&oldid=2741278 Corrected grammar and spelling errors - For this chapter - Sleep and ego depletion: How does sleep affect the capacity for self-control and willpower? - (Book chapter, 2024)] #11:14 pm, 11 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2020%2FChild_killer_motivation&diff=2832817&oldid=2721628 Used bullet points to organise see also section and provided few external sources - For this chapter - Child killer motivation: What motivates a child to kill? (Book chapter, 2020)] #9:02 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FBurnout&diff=2832924&oldid=2666399 Fixed overcapitalisation in focus questions, moved one reference into references section and fixed see also section - For this chapter - Burnout: What is burnout and how can be it managed and prevented - (Book chapter, 2022)] #9:37 pm, 12 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FTo-do_lists&diff=2832933&oldid=2655448 Corrected grammar and spelling errors - For this chapter -To-do lists: Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - (Book chapter, 2022)] #8:17 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAbusive_relationships_and_emotional_dependency&diff=2833144&oldid=2723675 Rewrote sentence to make it more clear to understand - For this chapter - Abusive relationships and emotional dependency: What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?(Book chapter, 2024)] #8:34 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FDopamine_fasting&diff=2833148&oldid=2716222 Rewrote few sentences to make it grammatically correct- For this chapter -Dopamine fasting: What is dopamine fasting and how does it affect motivation and emotion?- (Book chapter, 2024)] #8:41 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2018%2FBody_image_and_emotional_well-being&diff=2833150&oldid=2596066 Improved see also section - For this chapter - Body image and emotional well-being: How does body image influence one's emotional well being?- (Book chapter, 2018)] #8:47 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FClimate_change_helplessness&diff=2833152&oldid=2596067 Corrected few spellings and external links section - For this chapter -Climate change helplessness: How does learned helplessness impact motivation to engage in behaviours to limit climate change?(Book chapter, 2021)] #9:05 pm, 14 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FDental_fear&diff=2833154&oldid=2649741 Corrected grammatical errors - For this chapter -Dental fear: What causes dental fear, what are the consequences, and how can it be managed? - (Book chapter, 2021)] #7:01 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 Corrected 2 spelling mistake - For this chapter- Psychedelic treatment of eating disorders:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPsychedelic_treatment_of_eating_disorders&diff=2833299&oldid=2832896 How might psychedelic-assisted therapy influence psychological mechanisms involved in eating disorders? - (Book chapter, 2026)] #7:14 pm, 15 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FWisdom%2C_motivation%2C_and_emotion&diff=2833301&oldid=2831617 Corrected spelling error - For this chapter - Wisdom, motivation, and emotion: How do motivational and emotional processes contribute to wisdom? - (Book chapter, 2026)] #11:05 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FGender_and_achievement_motivation&diff=2833381&oldid=2832014 Corrected few grammatical errors - For this Chapter - Gender and achievement motivation: How does gender shape where, how, and under what conditions achievement motivation is expressed? - (Book chapter, 2026)] #11:23 am, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FVolunteer_wellbeing&diff=2833387&oldid=2829983 Corrected grammar error - For this chapter -Volunteer wellbeing: How does volunteering affect volunteers' subjective wellbeing? - (Book chapter, 2026)] #7: 38 pm, 16 September 2026: [[Talk:Motivation and emotion/Book/2026/Creative inspiration and effort|Commented on talk page and provided suggestion about theory that can be included in the chapter - For this chapter - Creative inspiration and effort: How do inspiration and effort interact during the creative process?]] #3:35 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 Corrected 2 spellings - For this chapter - Alcohol, dopamine, motivation, and emotion:] [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAlcohol%2C_dopamine%2C_motivation%2C_and_emotion&diff=2833572&oldid=2677570 What role does dopamine play in motivational and emotional responses to alcohol consumption? - (Book chapter, 2024)] #3:48 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FVasopressin_and_motivation&diff=2833579&oldid=2729665 Corrected spelling error - For this chapter - Vasopressin and motivation: How does vasopressin influence motivation? - (Book chapter, 2024)] #3:53 pm, 16 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FSleep_onset_optimisation&diff=2833580&oldid=2687478 Corrected 2 spellings - For this chapter - Sleep onset optimisation : What techniques assist in falling asleep promptly? - (Book chapter, 2024)] #2:12 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FProcrastination_and_temporal_discounting&diff=2833689&oldid=2761864 Corrected spelling errors - For this chapter - Procastination and temporal discounting: How does valuing immediate rewards over future outcomes contribute to procrastination? - (Book chapter, 2025)] #2:20 pm, 18 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FMaladaptive_defence_mechanisms_and_distress&diff=2833691&oldid=2761577 Corrected spelling errors - For this chapter- Maladaptive defence mechanisms and distress: In what ways do maladaptive defences shape emotional regulation and behavioural responses when under pressure? - (Book chapter, 2025)] #11:02 am, 19 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FSublimation&diff=2833828&oldid=2582187 Corrected 3 spelling errors - For this chapter -Sublimation: How can Sublimation be fostered? (Book chapter, 2023)] #1:50 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FInner_voice_in_anxiety&diff=2834960&oldid=2761762 Corrected spelling error - For this chapter- Inner voice in anxiety: What role does the inner voice play in maintaining or alleviating anxiety? - (Book chapter, 2025)] #4:52 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FMultiple_selves_theory%2C_motivation%2C_and_emotion&diff=2834969&oldid=2755077 Corrected grammar - For this chapter - Multiple selves, motivation, and emotion: How does the concept of multiple selves impact our motivational and emotional experiences? - (Book chapter, 2023)] #5:07 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FEmotional_eating&diff=2834971&oldid=2573916 Corrected spelling error - For this chapter - Emotional eating: What is emotional eating and what effect does it have on emotion and health?- (Book chapter, 2021)] #5:21 pm, 29 September 2026: [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2021%2FBrain_fog&diff=2834973&oldid=2345136 Corrected grammar - For this chapter - Brain fog: What is brain fog, what causes it, and how can it be managed?- (Book chapter, 2021)] 6p7yh8yr3effhwo5m91tzfjtgzgyk91 0 331713 2834952 2831819 2026-09-29T02:12:09Z James500 297601 /* Japanese */ Add 2834952 wikitext text/x-wiki {{Bibliography}} See also [[w:Bibliography of film by genre]], [[w:List of books on films]], [[w:Category:Books about film]] and [[s:Category:Works about film]] This part of the [[Universal Bibliography]] is a bibliography of cinema. Bibliography *George Rehrauer. The Macmillan Film Bibliography. [https://books.google.com/books?id=DmcoAAAAMAAJ] *Malte Hagener and Michael Töteberg. Film – An International Bibliography. Verlag JB Metzler. 2002. [https://books.google.co.uk/books?id=4r28DQAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Monaco and Susan Schenker. Books about Film: A Bibliographical Checklist. New York Zoetrope. 1976. [https://books.google.com/books?id=-xzgAAAAMAAJ] *Jack C Ellis, Charles Derry and Sharon Kern. The Film Book Bibliography, 1940-1975. Scarecrow Press. 1979. [https://books.google.com/books?id=BnWuAAAAIAAJ] *Robert A Armour. Film: A Reference Guide. 1980. [https://books.google.co.uk/books?id=8X5ZAAAAMAAJ] *Frank Manchel. Film Study: An Analytical Bibliography. 1990. [https://books.google.co.uk/books?id=BebEAji_wH4C&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=ID4E3Lm8TsgC&pg=PA953#v=onepage&q&f=false vol 2]. **Film Study: A Resource Guide. Fairleigh Dickinson University Press. 1973. [https://books.google.com/books?id=yzrgAAAAMAAJ] *The Film Index: A Bibliography. [https://books.google.co.uk/books?id=UM4SAQAAMAAJ] *Richard Dyer MacCann and Edward S Perry. The New Film Index: A Bibliography of Magazine Articles in English, 1930-1970. E P Dutton and Company. 1975. [https://books.google.com/books?id=FksOAQAAMAAJ] *Brian Reis. Australian Film: A Bibliography. Mansell. London and Washington. [https://books.google.co.uk/books?id=IP8PAQAAMAAJ] *Carl Vincent, Riccardo Redi and Franco Venturini (eds). Bibliografia Generale Del Cinema. [[w:it:Edizioni dell'Ateneo|Edizioni dell'Ateneo]]. Rome. 1953. **General Bibliography of Motion Pictures. Arno Press. [https://books.google.co.uk/books?id=qv1TAAAAMAAJ] General *Magill's Survey of Cinema Series *Screen Series. A Zwemmer Limited, London. A S Barnes & Co. New York. (eg Arne Svensson. Japan. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ]) *World Cinema. Flicks Books. [https://books.google.co.uk/books?id=Im9ZAAAAMAAJ vol 1]. [https://books.google.co.uk/books?id=7TxNAQAAIAAJ vol 2]. [https://books.google.co.uk/books?id=pS9AAQAAIAAJ vol 4]. [https://books.google.co.uk/books?id=AvjoDrhajOcC&pg=PP1#v=onepage&q&f=false vol 5]. [https://books.google.co.uk/books?id=yAInAQAAIAAJ] Periodicals See also [[w:List of film periodicals]] *International Index to Film Periodicals. [https://books.google.co.uk/books?id=0axkAAAAMAAJ 1974]. *Linda Batty. Retrospective Index to Film Periodicals, 1930-1971. R R Bowker Company. [https://books.google.co.uk/books?id=kcsUAQAAIAAJ] *East-West Film Journal [https://books.google.co.uk/books?id=3T8bAQAAIAAJ] (began December 1986) Annuals *Magill's Cinema Annual. [https://books.google.co.uk/books?id=LXAsctC3wasC 1983] (1982 films). [https://books.google.co.uk/books?id=V5XSO0pJmUUC 1986] (1985 films). *Picturegoer Film Annual. [https://books.google.co.uk/books?id=yhM6AQAAIAAJ for 1950-1951]. *The Motion Picture Annual. [https://books.google.co.uk/books?id=UhUIAQAAMAAJ 1989]. *Boy's Cinema Annual. Amalgamated Press. [https://books.google.co.uk/books?id=QVfych4dj54C 1939]. *International Film Annual [https://books.google.co.uk/books?id=D4I3AAAAIAAJ No 3] *The British Film Annual. Winchester Publications. [https://books.google.co.uk/books?id=QSMPAQAAMAAJ 1949]. *The Western Film Annual [https://books.google.co.uk/books?id=_kwoXXrpJU4C] Yearbooks *Film Daily Year Book of Motion Pictures. [https://books.google.co.uk/books?id=qJMHAQAAIAAJ 1969]. Years *Edgar Anstey, "The Cinema" (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes "Review of the Year". History *Arthur F McClure. Research Guide to Film History. R & E Publishers. 1983. [https://books.google.co.uk/books?id=iYcjAQAAIAAJ] Film reviews *Film Review Index. Oryx Press. **Volume 1: 1882-1949 [https://books.google.co.uk/books?id=EDngAAAAMAAJ] *Index to Critical Film Reviews in British and American Film Periodicals. Burt Franklin & Co. [https://books.google.co.uk/books?id=fM0UAQAAIAAJ] *The New York Times Film Reviews [https://books.google.co.uk/books?id=_E350jJ-Ui4C&pg=PP1#v=onepage&q&f=false] *[[w:Variety Film Reviews|Variety's Film Reviews]] [https://books.google.co.uk/books?id=62xZAAAAMAAJ] *Landers Film Reviews [https://books.google.co.uk/books?id=7d3jAAAAMAAJ] *[[w:Harrison's Reports and Film Reviews|Harrison's Reports and Film Reviews]] [https://books.google.co.uk/books?id=FFSDzDveXHgC] Writing *Timothy Corrigan. A Short Guide to Writing about Film. 2004. [https://books.google.co.uk/books?id=er8qAQAAIAAJ] Reading *James Monaco. How to Read a Film: The Art, Technology, Language, History, and Theory of Film and Media. 1981. [https://books.google.co.uk/books?id=inRZAAAAMAAJ&pg=PP1#v=onepage&q&f=false] World and national *Kevin Rockett and John Hill. National Cinemas and World Cinema. 2006. [https://books.google.co.uk/books?id=pudkAAAAMAAJ] World *Elkan Allan (comp). A Guide to World Cinema. Whittet Books. 1985. [https://books.google.co.uk/books?id=nb8dAAAAMAAJ] *Rob Stone, Paul Cooke, Stephanie Dennison and Alex Marlow-Mann (eds). The Routledge Companion to World Cinema. [https://books.google.co.uk/books?id=DnE3DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Geoffrey Nowell-Smith. The Oxford History of World Cinema. 1996. Paperback. 1997. [https://books.google.co.uk/books?id=MZwVDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *David Robinson. World Cinema: A Short History. Eyre Methuen. 1973. [https://books.google.co.uk/books?id=EH9ZAAAAMAAJ] *William Luhr (ed). World Cinema Since 1945. Ungar. New York. 1987. [https://books.google.co.uk/books?id=6gAqAAAAYAAJ] *James Chapman. Cinemas of the World: Film and Society from 1895 to the Present. 2003. [https://books.google.co.uk/books?id=SMYo4Abel2EC&pg=PP1#v=onepage&q&f=false] *Linda Badley (ed). Traditions in World Cinema. 2006. [https://books.google.co.uk/books?id=tpYkDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Kate Gamm. Teaching World Cinema. British Film Institute. 2004. [https://books.google.co.uk/books?id=PmJZAAAAMAAJ] *Shekhar Deshpande and Meta Mazaj. World Cinema: A Critical Introduction. 2018. [https://books.google.co.uk/books?id=qFFHDwAAQBAJ&pg=PA1#v=onepage&q&f=false] *John Hill and Pamela Church Gibson. World Cinema: Critical Approaches. 2000. [https://books.google.co.uk/books?id=SUKcAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lúcia Nagib, Chris Perriam and Rajinder Dudrah (eds). Theorizing World Cinema. 2012. [https://books.google.co.uk/books?id=cgNQEAAAQBAJ&pg=PP1#v=onepage&q&f=false] British and foreign *Jerry Vermilye (ed). 500 Best British and Foreign Films to Buy, Rent, Or Videotape. William Morrow. New York. 1988. [https://books.google.co.uk/books?id=91NZAAAAMAAJ] Foreign *Foreign Films: A Guide to Nearly 500 Films on Videocassette by Foreign Directors. CineBooks. 1989. [https://books.google.co.uk/books?id=NrtZAAAAYAAJ] *James Reid Paris. Classic Foreign Films: From 1960 to Today. Carol Publishing Group. 1993. [https://books.google.co.uk/books?id=PAQIAQAAMAAJ] *Michael F Mayer. Foreign Films on American Screens. Arco. 1965. [https://books.google.co.uk/books?id=-2ZZAAAAMAAJ] *Kerry Segrave. Foreign Films in America: A History. 2004. [https://books.google.co.uk/books?id=LVVp0omSDBIC&pg=PP1#v=onepage&q&f=false] Foreign language *Foreign Language Films. (Magill's Survey of Cinema). [https://books.google.co.uk/books?id=33xZAAAAMAAJ vol 1]. *Foreign Language Films. Visual Aids Service. Division of University Extension. University of Illinois. [https://books.google.co.uk/books?id=T_5PAQAAMAAJ 1964-1966]. [https://books.google.co.uk/books?id=Tv5PAQAAMAAJ 1970-1973]. *Michael S Barrett. Foreign Language Films and the Oscar: The Nominees and Winners, 1948-2017. McFarland & Company. 2018. [https://books.google.co.uk/books?id=P5NoDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Bibliography of Asian cinema *John A Lent. "Asian Cinema: A Selected International Bibliography". [https://books.google.co.uk/books?id=05RIAQAAIAAJ] **"Part I: East Asia, Featuring Japan" (1984) 36 Journal of Film and Video 75 (No 3: Summer 1984). "Japan" is at p 81. **"Part II: Southeast and South Asia, plus Taiwan and Korea" (1985) 37 Journal of Film and Video 76 (No 3: Spring 1985) Asian *Tom Vick. Asian Cinema: A Field Guide. HarperCollins. [https://books.google.co.uk/books?id=YpIaAQAAIAAJ] *Aaron Han Joon Magnan-Park, Gina Marchetti and See Kam Tan (eds). The Palgrave Handbook of Asian Cinema. 2018. [https://books.google.co.uk/books?id=MQ92DwAAQBAJ&pg=PR1#v=onepage&q&f=false] *Zhen Zhang, Sangjoon Lee, Debashree Mukherjee and Intan Paramaditha (eds). The Routledge Companion to Asian Cinemas. 2024. [https://books.google.co.uk/books?id=bwYDEQAAQBAJ&pg=PA1995#v=onepage&q&f=false] *Dimitris Eleftheriotis and Gary Needham. Asian Cinemas: A Reader and Guide. 2006. [https://books.google.co.uk/books?id=tPqgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Olivia Khoo. Asian Cinema: A Regional View. 2021. [https://books.google.co.uk/books?id=vvCgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stephen Teo. The Asian Cinema Experience: Styles, spaces, theory. Routledge. 2013. [https://books.google.co.uk/books?id=2sHZz-XUR2oC#v=onepage&q&f=false] *John A Lent. The Asian Film Industry. University of Texas Press. 1990. [https://books.google.co.uk/books?id=JkkqAAAAYAAJ] *Yau Shuk-ting and Kinnia (eds). East Asian Cinema and Cultural Heritage: From China, Hong Kong, Taiwan to Japan and South Korea. 2011. [https://books.google.co.uk/books?id=UeLHAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Routledge Handbook of South Asian Cinemas. 2026. [https://books.google.co.uk/books?id=QEyWEQAAQBAJ&pg=PA6#v=onepage&q&f=false] *Jose F Lacaba (ed). The Films of ASEAN. 2000. [https://books.google.co.uk/books?id=DI6dq64VNDkC] Periodicals, Asian cinema *Asian Cinema [https://books.google.co.uk/books?id=W4saAQAAIAAJ] *[[w:Cinemaya|Cinemaya]] [https://books.google.co.uk/books?id=3pMHAQAAIAAJ] *South Asian Cinema [https://books.google.co.uk/books?id=0eVkAAAAMAAJ] (began February 2001) Annuals, Asian cinema *Asian Film Directory & Who's Who. [https://books.google.co.uk/books?id=0wY6AQAAIAAJ 1952] India Hindi See [[w:Bibliography of Hindi cinema]] Japanese and Korean *Justin Bowyer. The Cinema of Japan & Korea. 2004. [https://books.google.co.uk/books?id=FtqUAxsisk0C&pg=PP1#v=onepage&q&f=false] *Kate E Taylor-Jones. Rising Sun, Divided Land: Japanese and South Korean Filmmakers. 2013. [https://books.google.co.uk/books?id=SfaEAAAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals, Japanese and Korean *Journal of Japanese and Korean Cinema [https://www.tandfonline.com/journals/rjkc20] ==Japanese== *Arne Svensson. Japan. (Screen Series). A Zwemmer Limited, London. A S Barnes & Co. New York. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ] *Daisuke Miyao (ed). The Oxford Handbook of Japanese Cinema. 2014. [https://books.google.co.uk/books?id=XXD1AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Joanne Bernardi, Shota T Ogawa (eds). Routledge Handbook of Japanese Cinema. 2021. [https://books.google.co.uk/books?id=xkX1DwAAQBAJ&pg=PA1931#v=onepage&q&f=false] *Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema: The Essential Handbook. [https://books.google.com/books?id=IrZcSAAACAAJ] *David Desser. A Companion to Japanese Cinema. 2022. [https://books.google.co.uk/books?id=YWl6EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hideaki Fujiki and Alastair Phillips (eds). The Japanese Cinema Book. Bloomsbury, for British Film Institute. 2020. [https://books.google.co.uk/books?id=Zu3cDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stuart Galbraith and Paul Duncan. Japanese Cinema. Taschen. 2009. ISBN 9783822831564. [https://books.google.co.uk/books?id=rDkFKAAACAAJ]. Catalogue: [https://catalogue.nla.gov.au/catalog/4701384] *"Japanese Cinema". Pam Cook (ed). The Cinema Book. British Film Institute. 1985. 3rd Ed: 2007. [https://books.google.co.uk/books?id=TTr8DwAAQBAJ&pg=PA238#v=onepage&q&f=false p 238]. *Martha P Nochimson. "Japan: Screening Feudalism and Modernism". World on Film: An Introduction. Chapter 5. [https://books.google.co.uk/books?id=DXK5gkQwAs8C&pg=PA195#v=onepage&q&f=false p 195]. *Donald Richie. Japanese Cinema: An Introduction. Oxford University Press. Oxford and New York. 1990. [https://books.google.co.uk/books?id=k4lZAAAAMAAJ]. Review: Cinemaya [https://books.google.co.uk/books?id=7ZEHAQAAIAAJ] *Marie Seton, "Japanese Cinema" (1958) [https://books.google.co.uk/books?id=Y5WQS46o0a4C 1] The Living Cinema 189 (No 4: Spring 1958) *Donald Richie. Japanese Cinema: Film Style and National Character. 1971. [https://books.google.com/books?id=0YlZAAAAMAAJ] *Joan Mellen. The Waves at Genji's Door: Japan Through Its Cinema. Pantheon Books. 1976. [https://books.google.co.uk/books?id=alC0AAAAIAAJ] *Joseph L Anderson and Donald Richie. The Japanese Film: Art and Industry. Tuttle. 1959. Evergreen. 1960. Expanded Edition. Princeton University Press. 1982. [https://books.google.co.uk/books?id=C2z3otM-y5kC&pg=PP1#v=onepage&q&f=false] *Stuart Galbraith IV. The Japanese Filmography: A Complete Reference to 209 Filmmakers and the Over 1250 Films Released in the United States, 1900 through 1994. 1996. [https://books.google.co.uk/books?id=GLxlEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *John Berra (ed). Directory of World Cinema: Japan. 2010. [https://books.google.co.uk/books?id=tQHsEAAAQBAJ&pg=PP1#v=onepage&q&f=false] **Directory of World Cinema: Japan 3. 2015. [https://books.google.co.uk/books?id=9QLsEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Abé Markus Nornes and Aaron Gerow. Research Guide to Japanese Film Studies. 2009. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Keiko I McDonald. Reading a Japanese Film: Cinema in Context. 2006. [https://books.google.co.uk/books?id=ICqSfjUqIpMC&pg=PP1#v=onepage&q&f=false] *David Bordwell, "Our Dream Cinema: Western Historiography and the Japanese Film" (1979) Film Reader, [https://books.google.co.uk/books?id=bp-RAAAAIAAJ No 4], p 45. *Mitsuyo Wada-Marciano. Japanese Cinema in the Digital Age. 2012. [https://books.google.co.uk/books?id=3F4EEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur Nolletti and David Desser. Reframing Japanese Cinema: Authorship, Genre, History. Indiana University Press. 1992. [https://books.google.co.uk/books?id=6Q1jAAAAMAAJ] *Laura Lee. Japanese Cinema Between Frames. 2017. [https://books.google.co.uk/books?id=c8I9DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Noel Burch. To the Distant Observer: Form and Meaning in the Japanese Cinema. 1979. [https://books.google.co.uk/books?id=lLtyz275-GYC&pg=PP1#v=onepage&q&f=false] *Dennis Washburn and Carole Cavanaugh (eds). Word and Image in Japanese Cinema. 2001. [https://books.google.co.uk/books?id=CwZ6SPpmJwQC&pg=PP1#v=onepage&q&f=false] *Daisuke Miyao. The Aesthetics of Shadow: Lighting and Japanese Cinema. 2013. [https://books.google.co.uk/books?id=80K2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Justin Vicari. Japanese Film and the Floating Mind: Cinematic Contemplations of Being. 2016. [https://books.google.co.uk/books?id=CYuwDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Keiko I McDonald. Cinema East: A Critical Study of Major Japanese Films. Fairleigh Dickinson University Press. Associated University Presses. 1983. [https://books.google.co.uk/books?id=ywxjAAAAMAAJ] *Alastair Phillips and Julian Stringer. Japanese Cinema: Texts and Contexts. 2007. [https://books.google.co.uk/books?id=53xRa5dwUOUC&pg=PP1#v=onepage&q&f=false] *Darrell William Davis. Picturing Japaneseness: Monumental Style, National Identity, Japanese Film. Columbia University Press. 1996. ISBN 0-231-10231-3. Review: [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/picturing-japaneseness-monumental-style-national-identity-japanese-film-by-darrell-william-davis-new-york-columbia-university-press-1996-viii-304-pp-1750/7CC73AD51C9A06CA40DF3463CC9D3E7D] *Barbara Wolf. The Japanese Film. Audio Brandon Films. Mount Vernon, NY. 1976. [https://books.google.co.uk/books?id=qo3SAAAAMAAJ] *[https://www.bfi.org.uk/articles/subject/japanese-cinema All articles about Japanese cinema]. BFI. Bibliography *Nornes and Gerow. "Annotated Bibliography for Bibliographic Studies". Research Guide to Japanese Film Studies. Chapter 4. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&pg=PA72#v=onepage&q&f=false p 72]. Reference *[https://guides.library.yale.edu/c.php?g=295932&p=1973050 Japanese Reference Materials for Studying Japanese Cinema at Yale University]. Yale Library. Periodicals See also [[w:ja:Category:日本の映画雑誌]]. [映画雑誌 = film magazine] [映画機関誌 = film journal] *Honchi Haruhiko (Japanese: 本地陽彦). Nihon Eiga Zasshi Taitoru Soran (Japanese: 日本映画雑誌タイトル総覧). [[w:ja:ワイズ出版|Waizu Shuppan]]. Tokyo. 2003. [https://books.google.co.uk/books?id=kskvAQAAIAAJ]. Commentary: Research Guide to Japanese Film Studies, [https://books.google.co.uk/books?id=5UONCwAAQBAJ&pg=PA83#v=onepage&q&f=false p 83]. *[https://www.nfaj.go.jp/exhibition/filmmagazines/ 映画雑誌の秘かな愉しみ The Discreet Charm of Film Magazines]. National Film Archive of Japan (NFAJ). *[https://guides.library.yale.edu/c.php?g=295932&p=1973061 Japanese Reference Materials for Studying Japanese Cinema at Yale University: Film Periodicals]. Yale Library. *[[w:en:Kinema Junpo|Kinema Junpo]] (Japanese: キネマ旬報) (The Movie Times) [https://books.google.co.uk/books?id=UFKtYBwc5ioC] *[[w:en:UniJapan Film Quarterly|UniJapan Film Quarterly]] [https://books.google.co.uk/books?id=QIY3AAAAIAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/AA12490478] Annuals and year books *Japanese Films. UniJapan Film. [https://books.google.co.uk/books?id=4LAaAQAAIAAJ 1960] **Japanese Film [https://books.google.co.uk/books?id=iJUHAQAAIAAJ 1983] [https://books.google.co.uk/books?id=fpYHAQAAIAAJ] *Cinema Year Book of Japan. [https://books.google.co.uk/books?id=r7W9DZnBbBwC 1938]. History *Jasper Sharp. Historical Dictionary of Japanese Cinema. 2011. [https://books.google.co.uk/books?id=YQR4EQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Isolde Standish. A New History of Japanese Cinema: A Century of Narrative Film. 2006. [https://books.google.co.uk/books?id=GZxUv4g_icQC&pg=PP1#v=onepage&q&f=false] *Donald Ritchie. A Hundred Years of Japanese Film: A Concise History, with a Selective Guide to DVDs and Videos. 2001. 2005. [https://books.google.co.uk/books?id=s7-_Gon5-a0C&pg=PP1#v=onepage&q&f=false] *Yomota Inuhiko. What Is Japanese Cinema? A History. 2014. 2019. [https://books.google.co.uk/books?id=8Il-DwAAQBAJ&pg=PA1#v=onepage&q&f=false] *Gerald Mast and Bruce F Kawin. "Japan" in "Cinemas East". A Short History of the Movies. Allyn and Bacon. 7th Ed: 2000: [https://books.google.co.uk/books?id=AC0IAQAAMAAJ]. pp 403 to 417. *Aaron Gerow. "From Misemono to Zigomar: A Discursive History of Early Japanese Cinema". Bean, Kapse and Horak (eds). Silent Cinema and the Politics of Space. 2014. Chapter 6. [https://books.google.co.uk/books?id=7poiAwAAQBAJ&pg=PA157#v=onepage&q&f=false p 157]. 20th century *Beverley Bare Buehrer. Japanese Films: A Filmography and Commentary, 1921-1989. McFarland & Company. 1990. [https://books.google.co.uk/books?id=N6RZAAAAMAAJ] *Aristides Gazetas. "Post-War Japanese Cinema: 1950-1990". An Introduction to World Cinema. 2nd Ed. 2008. Chapter 13. [https://books.google.co.uk/books?id=CPuZ-2UtVRwC&pg=PA169#v=onepage&q&f=false p 169]. *Japanese Experimental Film, 1960-1980. American Federation of Arts. [https://books.google.co.uk/books?id=XGUbAQAAIAAJ] *Yuriko Furuhata. Cinema of Actuality: Japanese Avant-Garde Filmmaking in the Season of Image Politics. 2013. [https://books.google.co.uk/books?id=sDi2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] [covers 1960s and 1970s]. *Outlaw Masters of Japanese Film. 2005. [https://books.google.co.uk/books?id=MbaKDwAAQBAJ&pg=PP1#v=onepage&q&f=false] [covers 1950s to 1970s] *Japanese Cinema: From Kurosawa To Tora-San" (1986) Information Bulletin, February 1986, p 15 [https://books.google.co.uk/books?id=Sw5PAQAAIAAJ] *Catherine Russell. Classical Japanese Cinema Revisited. 2011. [https://books.google.co.uk/books?id=uv9GAQAAQBAJ&pg=PR4#v=onepage&q&f=false] [covers roughly 1930 to 1960] *Kyoko Hirano. Mr. Smith Goes to Tokyo: Japanese Cinema Under the American Occupation, 1945-1952. 1992. [https://books.google.co.uk/books?id=6OsKAQAAMAAJ] *Peter B High. The Imperial Screen: Japanese Film Culture in the Fifteen Years' War, 1931-1945. 2003. [https://books.google.co.uk/books?id=6XiA9DOuvjAC&pg=PP1#v=onepage&q&f=false] *Naoki Yamamoto. Dialectics without Synthesis: Japanese Film Theory and Realism in a Global Frame. 2020. [https://books.google.co.uk/books?id=lEnrDwAAQBAJ&pg=PR1#v=onepage&q&f=false] [Covers 1910s to 1950s]. *Sean O'Reilly (ed). The Advent of Sound in Japanese Cinema: A Handbook. 2025. [https://books.google.co.uk/books?id=vAKbEQAAQBAJ&pg=PP1#v=onepage&q&f=false] [covers the 1930s] *Mitsuyo Wada-Marciano. Nippon Modern: Japanese Cinema of the 1920s and 1930s. 2008. [https://books.google.co.uk/books?id=dIc0RBYMs9kC&pg=PP1#v=onepage&q&f=false] *Iwamoto Kenji, "Japanese Cinema Until 1930: A Consideration of its Formal Aspects". Iris: A Journal of Theory on Image and Sound. No 16, p 9 [https://books.google.co.uk/books?id=zXFhCc7ENAEC] *David Bordwell, "Visual Style in Japanese Cinema, 1925-1945" (1995) [https://books.google.co.uk/books?id=YqIqAQAAIAAJ 7] Film History 5 to 31 (No 1: Spring 1995) *Sean D O'Reilly. Re-Viewing the Past: The Uses of History in the Cinema of Imperial Japan. 2018. Paperback Ed: 2020. [https://books.google.co.uk/books?id=fqFiDwAAQBAJ&pg=PP1#v=onepage&q&f=false] New, contemporary *Mark Schilling. Contemporary Japanese Film. 1999. [https://books.google.co.uk/books?id=LNQ4EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Kawamoto Saburo. New Trends in Japanese Cinema. (Orientation Seminars on Japan, No 21). The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=R5IqAQAAIAAJ] *Tadao Sato. Currents in Japanese Cinema. Kodansha International. 1982. [https://books.google.com/books?id=p-9kAAAAMAAJ] Genres *Abe Mark Nornes. Japanese Documentary Film: The Meiji Era Through Hiroshima. 2003. [https://books.google.co.uk/books?id=2JFrwF2LMw4C&pg=PP1#v=onepage&q&f=false] *S A Thornton. The Japanese Period Film: A Critical Analysis. 2008. [https://books.google.co.uk/books?id=yhPo95wPz8QC&pg=PP1#v=onepage&q&f=false] *Alain Silver. The Samurai Film. 1977: [https://books.google.co.uk/books?id=R4BZAAAAMAAJ]. The Overlook Press. 1983: [https://books.google.co.uk/books?id=AH1ZAAAAMAAJ] *David Desser. The Samurai Films of Akira Kurosawa. UMI Research Press. [https://books.google.com/books?id=AaJZAAAAMAAJ] *Silver Screen Samurai: The Best of Japan's Samurai Movie Posters. 2004. [https://books.google.co.uk/books?id=tdsuoae7X68C&pg=PA2#v=onepage&q&f=false] *Stuart Galbraith IV. Japanese Science Fiction, Fantasy and Horror Films. Macfarland. 1994. [https://books.google.co.uk/books?id=2XZREQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema Encyclopedia: The Horror, Fantasy, and SciFi films. Vital Books. 1997. [https://books.google.com/books?id=HGSybwAACAAJ] *Jay McRoy. Japanese Horror Cinema. University of Hawaii Press. Edinburgh University Press. 2005. [https://books.google.co.uk/books?id=XvOgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Review: Timothy Iles (2007) [https://books.google.co.uk/books?id=rt9KAQAAIAAJ 33] The Journal of Japanese Studies 264 *Colette Balmain. Introduction to Japanese Horror Film. 2008. [https://books.google.co.uk/books?id=POWqBgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Michael Crandol. Ghost in the Well: The Hidden History of Horror Films in Japan. 2021. [https://books.google.co.uk/books?id=mDgfEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Peter C Pugsley. Japanese High School Films: Iconography, Nostalgia and Discipline. 2022. [https://books.google.co.uk/books?id=jPOgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Peter C Pugsley. Beyond the High School Film: Reaching Adulthood in Everyday Japanese Cinema. 2026. [https://books.google.co.uk/books?id=7boEEgAAQBAJ&pg=PP1#v=onepage&q&f=false] Cult *Patrick Macias. Tokyoscope: The Japanese Cult Film Companion. 2001. [https://books.google.co.uk/books?id=g-0HAQAAMAAJ] Directors *Alexander Jacoby. A Critical Handbook of Japanese Film Directors: From the Silent Era to the Present Day. 2008. [https://books.google.co.uk/books?id=RhWNAgAAQBAJ&pg=PA1870#v=onepage&q&f=false] *Audie Bock. Japanese Film Directors. 1978. Paperback Ed. Kodansha International. 1985. [https://books.google.co.uk/books?id=A6EqAAAAYAAJ] *Mitsuhiro Yoshimoto. Kurosawa: Film Studies and Japanese Cinema. 2000. [https://books.google.co.uk/books?id=QizaCOjKs-IC&pg=PP1#v=onepage&q&f=false] Stars *Hideaki Fujiki. Making Personas: Transnational Film Stardom in Modern Japan. 2013. [https://books.google.co.uk/books?id=A_gFEAAAQBAJ&pg=PR1#v=onepage&q&f=false] Studios *Stuart Galbraith IV. The Toho Studios Story: A History and Complete Filmography. 2008. [https://books.google.co.uk/books?id=f7o8pq6G_dYC&pg=PP1#v=onepage&q&f=false] Interviews *Joan Mellen. Voices from the Japanese Cinema. Liveright. New York. 1975. ISBN 0871406047. [https://search.worldcat.org/title/Voices-from-the-Japanese-cinema/oclc/1245912150] V-Cinema *Tom Mes. Japanese Film and the Challenge of Video. 2023. [https://books.google.co.uk/books?id=70i7EAAAQBAJ&pg=PA1921#v=onepage&q&f=false] Film criticism *Scott Nygren. Time Frames: Japanese Cinema And the Unfolding of History. [https://books.google.co.uk/books?id=PvlgQgAACAAJ] [[Category:Film]] 6v01dyckrdra7lcjmjxljwghqieohof 2834953 2834952 2026-09-29T02:12:54Z James500 297601 /* Japanese */ Add 2834953 wikitext text/x-wiki {{Bibliography}} See also [[w:Bibliography of film by genre]], [[w:List of books on films]], [[w:Category:Books about film]] and [[s:Category:Works about film]] This part of the [[Universal Bibliography]] is a bibliography of cinema. Bibliography *George Rehrauer. The Macmillan Film Bibliography. [https://books.google.com/books?id=DmcoAAAAMAAJ] *Malte Hagener and Michael Töteberg. Film – An International Bibliography. Verlag JB Metzler. 2002. [https://books.google.co.uk/books?id=4r28DQAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Monaco and Susan Schenker. Books about Film: A Bibliographical Checklist. New York Zoetrope. 1976. [https://books.google.com/books?id=-xzgAAAAMAAJ] *Jack C Ellis, Charles Derry and Sharon Kern. The Film Book Bibliography, 1940-1975. Scarecrow Press. 1979. [https://books.google.com/books?id=BnWuAAAAIAAJ] *Robert A Armour. Film: A Reference Guide. 1980. [https://books.google.co.uk/books?id=8X5ZAAAAMAAJ] *Frank Manchel. Film Study: An Analytical Bibliography. 1990. [https://books.google.co.uk/books?id=BebEAji_wH4C&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=ID4E3Lm8TsgC&pg=PA953#v=onepage&q&f=false vol 2]. **Film Study: A Resource Guide. Fairleigh Dickinson University Press. 1973. [https://books.google.com/books?id=yzrgAAAAMAAJ] *The Film Index: A Bibliography. [https://books.google.co.uk/books?id=UM4SAQAAMAAJ] *Richard Dyer MacCann and Edward S Perry. The New Film Index: A Bibliography of Magazine Articles in English, 1930-1970. E P Dutton and Company. 1975. [https://books.google.com/books?id=FksOAQAAMAAJ] *Brian Reis. Australian Film: A Bibliography. Mansell. London and Washington. [https://books.google.co.uk/books?id=IP8PAQAAMAAJ] *Carl Vincent, Riccardo Redi and Franco Venturini (eds). Bibliografia Generale Del Cinema. [[w:it:Edizioni dell'Ateneo|Edizioni dell'Ateneo]]. Rome. 1953. **General Bibliography of Motion Pictures. Arno Press. [https://books.google.co.uk/books?id=qv1TAAAAMAAJ] General *Magill's Survey of Cinema Series *Screen Series. A Zwemmer Limited, London. A S Barnes & Co. New York. (eg Arne Svensson. Japan. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ]) *World Cinema. Flicks Books. [https://books.google.co.uk/books?id=Im9ZAAAAMAAJ vol 1]. [https://books.google.co.uk/books?id=7TxNAQAAIAAJ vol 2]. [https://books.google.co.uk/books?id=pS9AAQAAIAAJ vol 4]. [https://books.google.co.uk/books?id=AvjoDrhajOcC&pg=PP1#v=onepage&q&f=false vol 5]. [https://books.google.co.uk/books?id=yAInAQAAIAAJ] Periodicals See also [[w:List of film periodicals]] *International Index to Film Periodicals. [https://books.google.co.uk/books?id=0axkAAAAMAAJ 1974]. *Linda Batty. Retrospective Index to Film Periodicals, 1930-1971. R R Bowker Company. [https://books.google.co.uk/books?id=kcsUAQAAIAAJ] *East-West Film Journal [https://books.google.co.uk/books?id=3T8bAQAAIAAJ] (began December 1986) Annuals *Magill's Cinema Annual. [https://books.google.co.uk/books?id=LXAsctC3wasC 1983] (1982 films). [https://books.google.co.uk/books?id=V5XSO0pJmUUC 1986] (1985 films). *Picturegoer Film Annual. [https://books.google.co.uk/books?id=yhM6AQAAIAAJ for 1950-1951]. *The Motion Picture Annual. [https://books.google.co.uk/books?id=UhUIAQAAMAAJ 1989]. *Boy's Cinema Annual. Amalgamated Press. [https://books.google.co.uk/books?id=QVfych4dj54C 1939]. *International Film Annual [https://books.google.co.uk/books?id=D4I3AAAAIAAJ No 3] *The British Film Annual. Winchester Publications. [https://books.google.co.uk/books?id=QSMPAQAAMAAJ 1949]. *The Western Film Annual [https://books.google.co.uk/books?id=_kwoXXrpJU4C] Yearbooks *Film Daily Year Book of Motion Pictures. [https://books.google.co.uk/books?id=qJMHAQAAIAAJ 1969]. Years *Edgar Anstey, "The Cinema" (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes "Review of the Year". History *Arthur F McClure. Research Guide to Film History. R & E Publishers. 1983. [https://books.google.co.uk/books?id=iYcjAQAAIAAJ] Film reviews *Film Review Index. Oryx Press. **Volume 1: 1882-1949 [https://books.google.co.uk/books?id=EDngAAAAMAAJ] *Index to Critical Film Reviews in British and American Film Periodicals. Burt Franklin & Co. [https://books.google.co.uk/books?id=fM0UAQAAIAAJ] *The New York Times Film Reviews [https://books.google.co.uk/books?id=_E350jJ-Ui4C&pg=PP1#v=onepage&q&f=false] *[[w:Variety Film Reviews|Variety's Film Reviews]] [https://books.google.co.uk/books?id=62xZAAAAMAAJ] *Landers Film Reviews [https://books.google.co.uk/books?id=7d3jAAAAMAAJ] *[[w:Harrison's Reports and Film Reviews|Harrison's Reports and Film Reviews]] [https://books.google.co.uk/books?id=FFSDzDveXHgC] Writing *Timothy Corrigan. A Short Guide to Writing about Film. 2004. [https://books.google.co.uk/books?id=er8qAQAAIAAJ] Reading *James Monaco. How to Read a Film: The Art, Technology, Language, History, and Theory of Film and Media. 1981. [https://books.google.co.uk/books?id=inRZAAAAMAAJ&pg=PP1#v=onepage&q&f=false] World and national *Kevin Rockett and John Hill. National Cinemas and World Cinema. 2006. [https://books.google.co.uk/books?id=pudkAAAAMAAJ] World *Elkan Allan (comp). A Guide to World Cinema. Whittet Books. 1985. [https://books.google.co.uk/books?id=nb8dAAAAMAAJ] *Rob Stone, Paul Cooke, Stephanie Dennison and Alex Marlow-Mann (eds). The Routledge Companion to World Cinema. [https://books.google.co.uk/books?id=DnE3DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Geoffrey Nowell-Smith. The Oxford History of World Cinema. 1996. Paperback. 1997. [https://books.google.co.uk/books?id=MZwVDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *David Robinson. World Cinema: A Short History. Eyre Methuen. 1973. [https://books.google.co.uk/books?id=EH9ZAAAAMAAJ] *William Luhr (ed). World Cinema Since 1945. Ungar. New York. 1987. [https://books.google.co.uk/books?id=6gAqAAAAYAAJ] *James Chapman. Cinemas of the World: Film and Society from 1895 to the Present. 2003. [https://books.google.co.uk/books?id=SMYo4Abel2EC&pg=PP1#v=onepage&q&f=false] *Linda Badley (ed). Traditions in World Cinema. 2006. [https://books.google.co.uk/books?id=tpYkDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Kate Gamm. Teaching World Cinema. British Film Institute. 2004. [https://books.google.co.uk/books?id=PmJZAAAAMAAJ] *Shekhar Deshpande and Meta Mazaj. World Cinema: A Critical Introduction. 2018. [https://books.google.co.uk/books?id=qFFHDwAAQBAJ&pg=PA1#v=onepage&q&f=false] *John Hill and Pamela Church Gibson. World Cinema: Critical Approaches. 2000. [https://books.google.co.uk/books?id=SUKcAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lúcia Nagib, Chris Perriam and Rajinder Dudrah (eds). Theorizing World Cinema. 2012. [https://books.google.co.uk/books?id=cgNQEAAAQBAJ&pg=PP1#v=onepage&q&f=false] British and foreign *Jerry Vermilye (ed). 500 Best British and Foreign Films to Buy, Rent, Or Videotape. William Morrow. New York. 1988. [https://books.google.co.uk/books?id=91NZAAAAMAAJ] Foreign *Foreign Films: A Guide to Nearly 500 Films on Videocassette by Foreign Directors. CineBooks. 1989. [https://books.google.co.uk/books?id=NrtZAAAAYAAJ] *James Reid Paris. Classic Foreign Films: From 1960 to Today. Carol Publishing Group. 1993. [https://books.google.co.uk/books?id=PAQIAQAAMAAJ] *Michael F Mayer. Foreign Films on American Screens. Arco. 1965. [https://books.google.co.uk/books?id=-2ZZAAAAMAAJ] *Kerry Segrave. Foreign Films in America: A History. 2004. [https://books.google.co.uk/books?id=LVVp0omSDBIC&pg=PP1#v=onepage&q&f=false] Foreign language *Foreign Language Films. (Magill's Survey of Cinema). [https://books.google.co.uk/books?id=33xZAAAAMAAJ vol 1]. *Foreign Language Films. Visual Aids Service. Division of University Extension. University of Illinois. [https://books.google.co.uk/books?id=T_5PAQAAMAAJ 1964-1966]. [https://books.google.co.uk/books?id=Tv5PAQAAMAAJ 1970-1973]. *Michael S Barrett. Foreign Language Films and the Oscar: The Nominees and Winners, 1948-2017. McFarland & Company. 2018. [https://books.google.co.uk/books?id=P5NoDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Bibliography of Asian cinema *John A Lent. "Asian Cinema: A Selected International Bibliography". [https://books.google.co.uk/books?id=05RIAQAAIAAJ] **"Part I: East Asia, Featuring Japan" (1984) 36 Journal of Film and Video 75 (No 3: Summer 1984). "Japan" is at p 81. **"Part II: Southeast and South Asia, plus Taiwan and Korea" (1985) 37 Journal of Film and Video 76 (No 3: Spring 1985) Asian *Tom Vick. Asian Cinema: A Field Guide. HarperCollins. [https://books.google.co.uk/books?id=YpIaAQAAIAAJ] *Aaron Han Joon Magnan-Park, Gina Marchetti and See Kam Tan (eds). The Palgrave Handbook of Asian Cinema. 2018. [https://books.google.co.uk/books?id=MQ92DwAAQBAJ&pg=PR1#v=onepage&q&f=false] *Zhen Zhang, Sangjoon Lee, Debashree Mukherjee and Intan Paramaditha (eds). The Routledge Companion to Asian Cinemas. 2024. [https://books.google.co.uk/books?id=bwYDEQAAQBAJ&pg=PA1995#v=onepage&q&f=false] *Dimitris Eleftheriotis and Gary Needham. Asian Cinemas: A Reader and Guide. 2006. [https://books.google.co.uk/books?id=tPqgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Olivia Khoo. Asian Cinema: A Regional View. 2021. [https://books.google.co.uk/books?id=vvCgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stephen Teo. The Asian Cinema Experience: Styles, spaces, theory. Routledge. 2013. [https://books.google.co.uk/books?id=2sHZz-XUR2oC#v=onepage&q&f=false] *John A Lent. The Asian Film Industry. University of Texas Press. 1990. [https://books.google.co.uk/books?id=JkkqAAAAYAAJ] *Yau Shuk-ting and Kinnia (eds). East Asian Cinema and Cultural Heritage: From China, Hong Kong, Taiwan to Japan and South Korea. 2011. [https://books.google.co.uk/books?id=UeLHAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Routledge Handbook of South Asian Cinemas. 2026. [https://books.google.co.uk/books?id=QEyWEQAAQBAJ&pg=PA6#v=onepage&q&f=false] *Jose F Lacaba (ed). The Films of ASEAN. 2000. [https://books.google.co.uk/books?id=DI6dq64VNDkC] Periodicals, Asian cinema *Asian Cinema [https://books.google.co.uk/books?id=W4saAQAAIAAJ] *[[w:Cinemaya|Cinemaya]] [https://books.google.co.uk/books?id=3pMHAQAAIAAJ] *South Asian Cinema [https://books.google.co.uk/books?id=0eVkAAAAMAAJ] (began February 2001) Annuals, Asian cinema *Asian Film Directory & Who's Who. [https://books.google.co.uk/books?id=0wY6AQAAIAAJ 1952] India Hindi See [[w:Bibliography of Hindi cinema]] Japanese and Korean *Justin Bowyer. The Cinema of Japan & Korea. 2004. [https://books.google.co.uk/books?id=FtqUAxsisk0C&pg=PP1#v=onepage&q&f=false] *Kate E Taylor-Jones. Rising Sun, Divided Land: Japanese and South Korean Filmmakers. 2013. [https://books.google.co.uk/books?id=SfaEAAAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals, Japanese and Korean *Journal of Japanese and Korean Cinema [https://www.tandfonline.com/journals/rjkc20] ==Japanese== *Arne Svensson. Japan. (Screen Series). A Zwemmer Limited, London. A S Barnes & Co. New York. 1971. [https://books.google.co.uk/books?id=W4RZAAAAMAAJ] *Daisuke Miyao (ed). The Oxford Handbook of Japanese Cinema. 2014. [https://books.google.co.uk/books?id=XXD1AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Joanne Bernardi, Shota T Ogawa (eds). Routledge Handbook of Japanese Cinema. 2021. [https://books.google.co.uk/books?id=xkX1DwAAQBAJ&pg=PA1931#v=onepage&q&f=false] *Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema: The Essential Handbook. [https://books.google.com/books?id=IrZcSAAACAAJ] *David Desser. A Companion to Japanese Cinema. 2022. [https://books.google.co.uk/books?id=YWl6EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hideaki Fujiki and Alastair Phillips (eds). The Japanese Cinema Book. Bloomsbury, for British Film Institute. 2020. [https://books.google.co.uk/books?id=Zu3cDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stuart Galbraith and Paul Duncan. Japanese Cinema. Taschen. 2009. ISBN 9783822831564. [https://books.google.co.uk/books?id=rDkFKAAACAAJ]. Catalogue: [https://catalogue.nla.gov.au/catalog/4701384] *"Japanese Cinema". Pam Cook (ed). The Cinema Book. British Film Institute. 1985. 3rd Ed: 2007. [https://books.google.co.uk/books?id=TTr8DwAAQBAJ&pg=PA238#v=onepage&q&f=false p 238]. *Martha P Nochimson. "Japan: Screening Feudalism and Modernism". World on Film: An Introduction. Chapter 5. [https://books.google.co.uk/books?id=DXK5gkQwAs8C&pg=PA195#v=onepage&q&f=false p 195]. *Donald Richie. Japanese Cinema: An Introduction. Oxford University Press. Oxford and New York. 1990. [https://books.google.co.uk/books?id=k4lZAAAAMAAJ]. Review: Cinemaya [https://books.google.co.uk/books?id=7ZEHAQAAIAAJ] *Marie Seton, "Japanese Cinema" (1958) [https://books.google.co.uk/books?id=Y5WQS46o0a4C 1] The Living Cinema 189 (No 4: Spring 1958) *Donald Richie. Japanese Cinema: Film Style and National Character. 1971. [https://books.google.com/books?id=0YlZAAAAMAAJ] *Joan Mellen. The Waves at Genji's Door: Japan Through Its Cinema. Pantheon Books. 1976. [https://books.google.co.uk/books?id=alC0AAAAIAAJ] *Joseph L Anderson and Donald Richie. The Japanese Film: Art and Industry. Tuttle. 1959. Evergreen. 1960. Expanded Edition. Princeton University Press. 1982. [https://books.google.co.uk/books?id=C2z3otM-y5kC&pg=PP1#v=onepage&q&f=false] *Stuart Galbraith IV. The Japanese Filmography: A Complete Reference to 209 Filmmakers and the Over 1250 Films Released in the United States, 1900 through 1994. 1996. [https://books.google.co.uk/books?id=GLxlEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *John Berra (ed). Directory of World Cinema: Japan. 2010. [https://books.google.co.uk/books?id=tQHsEAAAQBAJ&pg=PP1#v=onepage&q&f=false] **Directory of World Cinema: Japan 3. 2015. [https://books.google.co.uk/books?id=9QLsEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Abé Markus Nornes and Aaron Gerow. Research Guide to Japanese Film Studies. 2009. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Keiko I McDonald. Reading a Japanese Film: Cinema in Context. 2006. [https://books.google.co.uk/books?id=ICqSfjUqIpMC&pg=PP1#v=onepage&q&f=false] *David Bordwell, "Our Dream Cinema: Western Historiography and the Japanese Film" (1979) Film Reader, [https://books.google.co.uk/books?id=bp-RAAAAIAAJ No 4], p 45. *Mitsuyo Wada-Marciano. Japanese Cinema in the Digital Age. 2012. [https://books.google.co.uk/books?id=3F4EEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur Nolletti and David Desser. Reframing Japanese Cinema: Authorship, Genre, History. Indiana University Press. 1992. [https://books.google.co.uk/books?id=6Q1jAAAAMAAJ] *Laura Lee. Japanese Cinema Between Frames. 2017. [https://books.google.co.uk/books?id=c8I9DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Noel Burch. To the Distant Observer: Form and Meaning in the Japanese Cinema. 1979. [https://books.google.co.uk/books?id=lLtyz275-GYC&pg=PP1#v=onepage&q&f=false] *Dennis Washburn and Carole Cavanaugh (eds). Word and Image in Japanese Cinema. 2001. [https://books.google.co.uk/books?id=CwZ6SPpmJwQC&pg=PP1#v=onepage&q&f=false] *Daisuke Miyao. The Aesthetics of Shadow: Lighting and Japanese Cinema. 2013. [https://books.google.co.uk/books?id=80K2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Justin Vicari. Japanese Film and the Floating Mind: Cinematic Contemplations of Being. 2016. [https://books.google.co.uk/books?id=CYuwDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Keiko I McDonald. Cinema East: A Critical Study of Major Japanese Films. Fairleigh Dickinson University Press. Associated University Presses. 1983. [https://books.google.co.uk/books?id=ywxjAAAAMAAJ] *Alastair Phillips and Julian Stringer. Japanese Cinema: Texts and Contexts. 2007. [https://books.google.co.uk/books?id=53xRa5dwUOUC&pg=PP1#v=onepage&q&f=false] *Darrell William Davis. Picturing Japaneseness: Monumental Style, National Identity, Japanese Film. Columbia University Press. 1996. ISBN 0-231-10231-3. Review: [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/picturing-japaneseness-monumental-style-national-identity-japanese-film-by-darrell-william-davis-new-york-columbia-university-press-1996-viii-304-pp-1750/7CC73AD51C9A06CA40DF3463CC9D3E7D] *Barbara Wolf. The Japanese Film. Audio Brandon Films. Mount Vernon, NY. 1976. [https://books.google.co.uk/books?id=qo3SAAAAMAAJ] *[https://www.bfi.org.uk/articles/subject/japanese-cinema All articles about Japanese cinema]. BFI. Bibliography *Nornes and Gerow. "Annotated Bibliography for Bibliographic Studies". Research Guide to Japanese Film Studies. Chapter 4. [https://books.google.co.uk/books?id=5UONCwAAQBAJ&pg=PA72#v=onepage&q&f=false p 72]. Reference *[https://guides.library.yale.edu/c.php?g=295932&p=1973050 Japanese Reference Materials for Studying Japanese Cinema at Yale University]. Yale Library. Periodicals See also [[w:ja:Category:日本の映画雑誌]]. [映画雑誌 = film magazine] [映画機関誌 = film journal] *Honchi Haruhiko (Japanese: 本地陽彦). Nihon Eiga Zasshi Taitoru Soran (Japanese: 日本映画雑誌タイトル総覧). [[w:ja:ワイズ出版|Waizu Shuppan]]. Tokyo. 2003. [https://books.google.co.uk/books?id=kskvAQAAIAAJ]. Commentary: Research Guide to Japanese Film Studies, [https://books.google.co.uk/books?id=5UONCwAAQBAJ&pg=PA83#v=onepage&q&f=false p 83]. *[https://www.nfaj.go.jp/exhibition/filmmagazines/ 映画雑誌の秘かな愉しみ The Discreet Charm of Film Magazines]. National Film Archive of Japan (NFAJ). *[https://guides.library.yale.edu/c.php?g=295932&p=1973061 Japanese Reference Materials for Studying Japanese Cinema at Yale University: Film Periodicals]. Yale Library. *[[w:en:Kinema Junpo|Kinema Junpo]] (Japanese: キネマ旬報) (The Movie Times) [https://books.google.co.uk/books?id=UFKtYBwc5ioC] *[[w:en:UniJapan Film Quarterly|UniJapan Film Quarterly]] [https://books.google.co.uk/books?id=QIY3AAAAIAAJ]. Catalogue: [https://ci.nii.ac.jp/ncid/AA12490478] Annuals and year books *Japanese Films. UniJapan Film. [https://books.google.co.uk/books?id=4LAaAQAAIAAJ 1960] **Japanese Film [https://books.google.co.uk/books?id=iJUHAQAAIAAJ 1983] [https://books.google.co.uk/books?id=fpYHAQAAIAAJ] *Cinema Year Book of Japan. [https://books.google.co.uk/books?id=r7W9DZnBbBwC 1938]. History *Jasper Sharp. Historical Dictionary of Japanese Cinema. 2011. [https://books.google.co.uk/books?id=YQR4EQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Isolde Standish. A New History of Japanese Cinema: A Century of Narrative Film. 2006. [https://books.google.co.uk/books?id=GZxUv4g_icQC&pg=PP1#v=onepage&q&f=false] *Donald Ritchie. A Hundred Years of Japanese Film: A Concise History, with a Selective Guide to DVDs and Videos. 2001. 2005. [https://books.google.co.uk/books?id=s7-_Gon5-a0C&pg=PP1#v=onepage&q&f=false] *Yomota Inuhiko. What Is Japanese Cinema? A History. 2014. 2019. [https://books.google.co.uk/books?id=8Il-DwAAQBAJ&pg=PA1#v=onepage&q&f=false] *Gerald Mast and Bruce F Kawin. "Japan" in "Cinemas East". A Short History of the Movies. Allyn and Bacon. 7th Ed: 2000: [https://books.google.co.uk/books?id=AC0IAQAAMAAJ]. pp 403 to 417. *Aaron Gerow. "From Misemono to Zigomar: A Discursive History of Early Japanese Cinema". Bean, Kapse and Horak (eds). Silent Cinema and the Politics of Space. 2014. Chapter 6. [https://books.google.co.uk/books?id=7poiAwAAQBAJ&pg=PA157#v=onepage&q&f=false p 157]. 20th century *Beverley Bare Buehrer. Japanese Films: A Filmography and Commentary, 1921-1989. McFarland & Company. 1990. [https://books.google.co.uk/books?id=N6RZAAAAMAAJ] *Aristides Gazetas. "Post-War Japanese Cinema: 1950-1990". An Introduction to World Cinema. 2nd Ed. 2008. Chapter 13. [https://books.google.co.uk/books?id=CPuZ-2UtVRwC&pg=PA169#v=onepage&q&f=false p 169]. *Japanese Experimental Film, 1960-1980. American Federation of Arts. [https://books.google.co.uk/books?id=XGUbAQAAIAAJ] *Yuriko Furuhata. Cinema of Actuality: Japanese Avant-Garde Filmmaking in the Season of Image Politics. 2013. [https://books.google.co.uk/books?id=sDi2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] [covers 1960s and 1970s]. *Outlaw Masters of Japanese Film. 2005. [https://books.google.co.uk/books?id=MbaKDwAAQBAJ&pg=PP1#v=onepage&q&f=false] [covers 1950s to 1970s] *Japanese Cinema: From Kurosawa To Tora-San" (1986) Information Bulletin, February 1986, p 15 [https://books.google.co.uk/books?id=Sw5PAQAAIAAJ] *Catherine Russell. Classical Japanese Cinema Revisited. 2011. [https://books.google.co.uk/books?id=uv9GAQAAQBAJ&pg=PR4#v=onepage&q&f=false] [covers roughly 1930 to 1960] *Matthew Thrift. [https://www.bfi.org.uk/lists/10-great-japanese-films-1950s 10 great Japanese films of the 1950s]. BFI. 22 August 2024. *Kyoko Hirano. Mr. Smith Goes to Tokyo: Japanese Cinema Under the American Occupation, 1945-1952. 1992. [https://books.google.co.uk/books?id=6OsKAQAAMAAJ] *Peter B High. The Imperial Screen: Japanese Film Culture in the Fifteen Years' War, 1931-1945. 2003. [https://books.google.co.uk/books?id=6XiA9DOuvjAC&pg=PP1#v=onepage&q&f=false] *Naoki Yamamoto. Dialectics without Synthesis: Japanese Film Theory and Realism in a Global Frame. 2020. [https://books.google.co.uk/books?id=lEnrDwAAQBAJ&pg=PR1#v=onepage&q&f=false] [Covers 1910s to 1950s]. *Sean O'Reilly (ed). The Advent of Sound in Japanese Cinema: A Handbook. 2025. [https://books.google.co.uk/books?id=vAKbEQAAQBAJ&pg=PP1#v=onepage&q&f=false] [covers the 1930s] *Mitsuyo Wada-Marciano. Nippon Modern: Japanese Cinema of the 1920s and 1930s. 2008. [https://books.google.co.uk/books?id=dIc0RBYMs9kC&pg=PP1#v=onepage&q&f=false] *Iwamoto Kenji, "Japanese Cinema Until 1930: A Consideration of its Formal Aspects". Iris: A Journal of Theory on Image and Sound. No 16, p 9 [https://books.google.co.uk/books?id=zXFhCc7ENAEC] *David Bordwell, "Visual Style in Japanese Cinema, 1925-1945" (1995) [https://books.google.co.uk/books?id=YqIqAQAAIAAJ 7] Film History 5 to 31 (No 1: Spring 1995) *Sean D O'Reilly. Re-Viewing the Past: The Uses of History in the Cinema of Imperial Japan. 2018. Paperback Ed: 2020. [https://books.google.co.uk/books?id=fqFiDwAAQBAJ&pg=PP1#v=onepage&q&f=false] New, contemporary *Mark Schilling. Contemporary Japanese Film. 1999. [https://books.google.co.uk/books?id=LNQ4EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Kawamoto Saburo. New Trends in Japanese Cinema. (Orientation Seminars on Japan, No 21). The Japan Foundation, Office for the Japanese Studies Center. [https://books.google.co.uk/books?id=R5IqAQAAIAAJ] *Tadao Sato. Currents in Japanese Cinema. Kodansha International. 1982. [https://books.google.com/books?id=p-9kAAAAMAAJ] Genres *Abe Mark Nornes. Japanese Documentary Film: The Meiji Era Through Hiroshima. 2003. [https://books.google.co.uk/books?id=2JFrwF2LMw4C&pg=PP1#v=onepage&q&f=false] *S A Thornton. The Japanese Period Film: A Critical Analysis. 2008. [https://books.google.co.uk/books?id=yhPo95wPz8QC&pg=PP1#v=onepage&q&f=false] *Alain Silver. The Samurai Film. 1977: [https://books.google.co.uk/books?id=R4BZAAAAMAAJ]. The Overlook Press. 1983: [https://books.google.co.uk/books?id=AH1ZAAAAMAAJ] *David Desser. The Samurai Films of Akira Kurosawa. UMI Research Press. [https://books.google.com/books?id=AaJZAAAAMAAJ] *Silver Screen Samurai: The Best of Japan's Samurai Movie Posters. 2004. [https://books.google.co.uk/books?id=tdsuoae7X68C&pg=PA2#v=onepage&q&f=false] *Stuart Galbraith IV. Japanese Science Fiction, Fantasy and Horror Films. Macfarland. 1994. [https://books.google.co.uk/books?id=2XZREQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Thomas Weisser and Yuko Mihara Weisser. Japanese Cinema Encyclopedia: The Horror, Fantasy, and SciFi films. Vital Books. 1997. [https://books.google.com/books?id=HGSybwAACAAJ] *Jay McRoy. Japanese Horror Cinema. University of Hawaii Press. Edinburgh University Press. 2005. [https://books.google.co.uk/books?id=XvOgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Review: Timothy Iles (2007) [https://books.google.co.uk/books?id=rt9KAQAAIAAJ 33] The Journal of Japanese Studies 264 *Colette Balmain. Introduction to Japanese Horror Film. 2008. [https://books.google.co.uk/books?id=POWqBgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Michael Crandol. Ghost in the Well: The Hidden History of Horror Films in Japan. 2021. [https://books.google.co.uk/books?id=mDgfEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Peter C Pugsley. Japanese High School Films: Iconography, Nostalgia and Discipline. 2022. [https://books.google.co.uk/books?id=jPOgEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Peter C Pugsley. Beyond the High School Film: Reaching Adulthood in Everyday Japanese Cinema. 2026. [https://books.google.co.uk/books?id=7boEEgAAQBAJ&pg=PP1#v=onepage&q&f=false] Cult *Patrick Macias. Tokyoscope: The Japanese Cult Film Companion. 2001. [https://books.google.co.uk/books?id=g-0HAQAAMAAJ] Directors *Alexander Jacoby. A Critical Handbook of Japanese Film Directors: From the Silent Era to the Present Day. 2008. [https://books.google.co.uk/books?id=RhWNAgAAQBAJ&pg=PA1870#v=onepage&q&f=false] *Audie Bock. Japanese Film Directors. 1978. Paperback Ed. Kodansha International. 1985. [https://books.google.co.uk/books?id=A6EqAAAAYAAJ] *Mitsuhiro Yoshimoto. Kurosawa: Film Studies and Japanese Cinema. 2000. [https://books.google.co.uk/books?id=QizaCOjKs-IC&pg=PP1#v=onepage&q&f=false] Stars *Hideaki Fujiki. Making Personas: Transnational Film Stardom in Modern Japan. 2013. [https://books.google.co.uk/books?id=A_gFEAAAQBAJ&pg=PR1#v=onepage&q&f=false] Studios *Stuart Galbraith IV. The Toho Studios Story: A History and Complete Filmography. 2008. [https://books.google.co.uk/books?id=f7o8pq6G_dYC&pg=PP1#v=onepage&q&f=false] Interviews *Joan Mellen. Voices from the Japanese Cinema. Liveright. New York. 1975. ISBN 0871406047. [https://search.worldcat.org/title/Voices-from-the-Japanese-cinema/oclc/1245912150] V-Cinema *Tom Mes. Japanese Film and the Challenge of Video. 2023. [https://books.google.co.uk/books?id=70i7EAAAQBAJ&pg=PA1921#v=onepage&q&f=false] Film criticism *Scott Nygren. Time Frames: Japanese Cinema And the Unfolding of History. [https://books.google.co.uk/books?id=PvlgQgAACAAJ] [[Category:Film]] gm3qyg4y1n0emdvfwt88g25plipe6tr 0 331726 2834965 2829814 2026-09-29T04:45:17Z Jtneill 10242 Copyediting 2834965 wikitext text/x-wiki {{title|Deliberative vs implemental mindset:<br>What are the motivational and cognitive differences between deliberative and implemental mindsets?}} == Overview == The deliberative and implemental mindset are two distinct cognitive processes associated with differing stages in the pursuit of a goal. The distinction of these mindsets was developed and coined by psychologist Peter Gollwitzer during his work on the Rubicon of Action theory {{fact}} and later superseded by the Mindsets in the Control of Action theory developed later on {{fact}}. The deliberative mindset is primarily associated with deciding which goal to pursue, whilst the implemental mindset is associated with the how and how to achieve proposed goal. == '''Background''' == According to Gollwitzer {{fact}}, goal directed action involved four distinct stages, including deciding what they as an individual want to achieve, deciding how an individual will achieve said goal and assessing the outcome of achieving proposed goal. Gollwitzer proposed that differing stages involved in this process created differing psychological demands and activated different cognitive processes, or mindsets, to meet these demands. The distinction Gollwitzer made originated from researching the difference between pre-decisional and post-decisional stages in which an individual is made to consider their goal, it's desirability and feasibility, as it considers that desires and needs to be beyond what is achievable in a lifetime. After a decision has been made attention or mindset will shift towards the necessary steps need to achieve said goal, or planning and implementation. === Mindset of Theory of Action === The mindset of theory of action is separated into four stages that can be associated with both the deliberative and implementation activated across the four stages of goal pursuit {{fact}} {{ic|Use bullet points as shown in Tutorail 2}} '''Pre-decisional phase''' '''-''' deliberating on a goal, weighing up feasibility and desirability '''Post-decisional phase''' '''-''' implementation of action on a goal '''Actional Phase''' '''-''' carrying out sustained action to achieve said goal '''Post- Actional phase''' '''-''' assessing out outcome of desired and reached goal The mindset of action perfectly demonstrates the deliberative and implementation mindset in action when undertaking goal directed behaviour. === '''Deliberative Mindset''' === A deliberative mindset occurs during the pre-decisional stage of pursuing a goal, an individual is weighing up desirability and feasibility and have not committed themselves to a goal to pursue. ==== '''Characteristics of a deliberative mindset''' ==== * Consideration of multiple options, pathways and alternatives * Consider desirability and feasibility of proposed goals * Be realistic of likelihood of success of goal * Take consideration of available information in support or contradiction of goal A deliberative mindset can be summed in a nutshell when considering this scenario, an individual is freshly eighteen years old, they have completed their HSC, so what's next? Do they pursue vocational study, do a gap year, pursue full-time employment or pursue university. In this scenario, an individual is entering the pre-decisional phase or activating their deliberative mindset, weighing up cost,duration, benefits, potential career pathways and likelihood of success which is itself based on aptitude and inclination. So the question is, what's next? === '''Implementation Mindset''' === What's next occurs after an individual has committed to their pursuit of a goal and begins planning how pursue that goal and the necessary steps needed to achieve that goal. Therefore it is associated with the post decisional stage of goal pursuit. The individual has moved on and deliberated on the feasibility and the likelihood of success of the proposed goal. The individual will then focus cognitive resources on how to achieve said goal. So the next question is, how do i achieve my goal? ==== '''Characteristics of an Implementation mindset''' ==== * Focus on the when, why, where and how * Identify the actions needed to achieve settled on goal * Anticipate and develop strategies to overcome obstacles and potential difficulties Consider the earlier scenario {{huh}}, an individual is freshly eighteen, they have completed their HSC, so the pathways become vocational study (apprenticeship), university, gap year or full-time employment. They've identified that they would pursue psychology, so what's next for them? They decide which university, weighing potential cost, reputation, they determine study/life to achieve the best grades without burning out, where psychology will take them, will it be human resources, marketing, recruiting or healthcare. Psychology can be applied anywhere it is an incredibly broad discipline. === Comparison === {{ic|Add an APA style table caption}} {| class="wikitable" ! colspan="undefined" |Feature ! colspan="undefined" |Deliberative Mindset ! colspan="undefined" |Implemental Mindset |- | colspan="undefined" |'''Action Phase''' | colspan="undefined" |Pre-decisional (Goal Setting) | colspan="undefined" |Post-decisional (Goal Striving) |- | colspan="undefined" |'''Cognitive Style''' | colspan="undefined" |Open-minded, impartial | colspan="undefined" |Closed-minded, partial |- | colspan="undefined" |'''Information Focus''' | colspan="undefined" |Feasibility and desirability pros/cons | colspan="undefined" |Execution steps and "how-to" plans |- | colspan="undefined" |'''Attention Span''' | colspan="undefined" |Broad visual and mental attention | colspan="undefined" |Narrowed, focused on foreground tasks |} {{ic|What is the source for the information contained in this table?}} === Limitations === The distinction between the deliberative and implementation mindset exhibits that in pursuit of your own individual goals, differentiating mindsets are essential in the pursuit of a goal. In a deliberative mindset, an individual considers their options, the steps that it<nowiki>''</nowiki>ll take to reach goal, whereas an implementation mindset will activate when a decision has been reached and requires sustained long-term action. However, it's important to consider an excessively long sustained deliberative mindset may lead to indecisiveness, procrastination and difficulty taking action. Whilst, an implementation mindset elicited too hastily may lead to an ill-informed choice or an undesirable or unfeasible outcome. It should be consider that each individual is different and mindsets change as new information is received either substantially increasing desirability and feasibility or having the opposite effect. An individual may decide on the university pathway, as opposed to vocational study but lose out on long-term financial gains, if they were to do an apprenticeship or traineeship, they would have to consider job security and whether technological advancement can affect their future career. == '''References''' == {{Hanging indent|1= Keller, L., Gollwitzer, P., & Sheeran, P. (n.d.). ''6 Changing Behavior Using the Model of Action Phases''. Retrieved August 31, 2026, from https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/c/6235/files/2020/08/Keller-Gollwitzer-Sheeran-Changing_behavior_using_the_model_of_action_phases.pdf?bid=6235 Gollwitzer, P. M., & Bayer, U. (2026). ''Deliberative versus implemental mindsets in the control of action | Social Psychology & Motivation''. Uni-Konstanz.De. https://www.socmot.uni-konstanz.de/publications/deliberative-versus-implemental-mindsets-control-action?utm_ Brandstatter, V. (2015, March). ''(PDF) the role of deliberative versus implemental mindsets in time prediction and task accomplishment''. ResearchGate. https://www.researchgate.net/publication/275632731_The_Role_of_Deliberative_Versus_Implemental_Mindsets_in_Time_Prediction_and_Task_Accomplishment Keller, L., Bieleke, M., & Gollwitzer, P. M. (2019). Mindset theory of action phases and IF-then planning. ''Social Psychology in Action'', 23–37. https://doi.org/10.1007/978-3-030-13788-5_2 }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Mindset]] k5xk94e4jqh0h2h47llaoim667qjote 1 331836 2834991 2832299 2026-09-29T07:47:46Z U3239236 3106753 /* Heading casing */ Reply 2834991 wikitext text/x-wiki == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:ChelsSchofield|Hi [[User:ChelsSchofield|ChelsSchofield]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:24, 5 September 2026 (UTC) |} :Thank you for the detailed marking. I will actively work towards these improvements. [[User:ChelsSchofield|ChelsSchofield]] ([[User talk:ChelsSchofield|discuss]] • [[Special:Contributions/ChelsSchofield|contribs]]) 00:58, 9 September 2026 (UTC) ::Hi! I really like how you've connected pain perception with different technologies and psychological theories. I noticed that parts of the “Understanding pain perception and management” section still read like planning notes, for example, “Relate this distinction to technology” and “Refer to increased attention towards bodily sensations.” It might help to turn these into complete paragraphs explaining the ideas directly to the reader. I think this would make the section flow more smoothly and clarify the connection between the psychological theory and technology. Hope this helps! [[User:U3239236|U3239236]] ([[User talk:U3239236|discuss]] • [[Special:Contributions/U3239236|contribs]]) 07:47, 29 September 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement <!-- Alignment with focus questions --> # Develop closer alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Good <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image (not a diagram) to the scenario to help attract reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Solid development # Include an introductory paragraph before branching into sub-sections <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. ... (fixed) <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section <!-- Tables --> # Promising use of table(s) |7= <!-- References --> <!-- Overall --> # Basic <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## [[Help:Wikitext quick reference|italicisation]] ## include hyperlinked dois |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Very good <!-- Description about self --> # Excellent description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |10= <!-- Social contribution --> # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:24, 5 September 2026 (UTC) d2htd84f4tm3phjbezefranlkdin470 0 331984 2835021 2832741 2026-09-29T10:33:37Z The Citer 3110681 /* Classes */ 2835021 wikitext text/x-wiki [[Image:Lunularia cruciata.jpg|thumb|300px|right|This is a Liverwort.]] Marchantiophytes are non-vascular land plants in the division Marchantiophyta, also known as hepatics. ==Information== Name Meaning: Marchantia-like plant, liver plant English Common Name: Liverworts Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system Approximate number of species described: 9,000 ==Classes== Haplomitriopsida – Jungermanniopsida – Marchantiopsida ==Evolutionary History== [https://en.wikipedia.org/wiki/Epiphyte Epiphytic] thalloid liverworts evolved during the Triassic to the Cretaceous. ==References== [[Wikipedia:Liverwort]] a72mpf00cjxdlxcbfty4yv7k36jgqrb 2835031 2835021 2026-09-29T10:52:13Z The Citer 3110681 Added phylogeny. 2835031 wikitext text/x-wiki [[Image:Lunularia cruciata.jpg|thumb|300px|right|This is a Liverwort.]] Marchantiophytes are non-vascular land plants in the division Marchantiophyta, also known as hepatics. ==Information== Name Meaning: Marchantia-like plant, liver plant English Common Name: Liverworts Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system Approximate number of species described: 9,000 ==Classes== Haplomitriopsida – Jungermanniopsida – Marchantiopsida ==Evolutionary History== [https://en.wikipedia.org/wiki/Epiphyte Epiphytic] thalloid liverworts evolved during the Triassic to the Cretaceous. ==Phylogeny== {{Clade | style=font-size:90%;line-height:80% |1={{clade |1={{clade |label1=Haplomitriopsida |1={{clade |1=Haplomitriales |2=Treubiales }} }} |2={{clade |label1=Marchantiopsida |1={{clade |label1=Blasiidae |1=Blasiales |label2=Marchantiidae |2={{clade |1=Neohodgsoniales |2={{clade |1=Sphaerocarpales |2={{clade |1=Lunulariales |2=Marchantiales }} }} }} }} |label2=Jungermanniopsida |2={{clade |1={{clade |label1=Pelliidae |1={{clade |1=Pelliales |2={{clade |1=Fossombroniales |2=Pallaviciniales }} }} }} |2={{clade |label1=Metzgeriidae |1={{clade |1=Metzgeriales |2=Pleuroziales }} |label2=Jungermanniidae |2={{clade |1=Jungermanniales |2={{clade |1=Porellales |2=Ptilidiales }} }} }} }} }} }} }} ==References== [[Wikipedia:Liverwort]] 7qigcl4ipi3h9olj2z6j2nl3cetwsfx 0 331998 2834855 2833591 2026-09-28T14:43:45Z Evgenii Arsentev 3110906 Add the 722-session log study (v1.1, corrected costs) to section 1 and references; cite contextburn; remove the superseded 12.5x cache ratio 2834855 wikitext text/x-wiki {{Short description|Why agent runs cost what they do, and how to measure it}} This learning resource is a sub-page of [[Large language models]]. It is about a practical question that anyone running a coding agent meets within a week of starting: '''why does a session that does ordinary work cost what it costs, and what actually reduces that cost?''' The short answer is that the cost of an agent run is dominated not by the text the model writes, but by the text it reads again and again. This resource explains the mechanism, shows how to measure it without the two mistakes that make the measurement wrong, and works through a controlled experiment whose result contradicts the advice most commonly given. {{RightTOC}} == Learning objectives == After working through this resource a learner should be able to: * explain why the resource consumption of a multi-step agent run grows faster than the amount of text the agent produces; * distinguish '''cache write''' from '''cache read''' and say why their price difference determines the shape of the cost curve; * take a usage log from an agent runtime and compute the cost of a run correctly, avoiding double counting and truncated records; * design a controlled comparison between two ways of working with an agent, and state honestly what such a comparison can and cannot establish; * read a U-shaped cost curve and recognise when the bottom of it is a ''plateau'' rather than a point. == 1. Why repeated input dominates == An agent that works on a task over several steps does not send a fresh question to the model each time. It sends the accumulated conversation: the original instructions, every file it has read, every command it has run, and every answer it has produced so far. Then it appends the next step and sends the whole thing again. The consequence is worth stating as an inequality. If a run has ''n'' steps and the context after step ''i'' has length ''c_i'', the total input the model processes is : <math>\sum_{i=1}^{n} c_i</math> while the total output is only the sum of what was generated at each step, which in practical runs is one to two orders of magnitude smaller. A token admitted into the context on the first step of a sixty-step run is paid for sixty times. A token generated on the last step is paid for once. This is why intuitions carried over from single-turn use of a language model mislead. In single-turn use, "keep the answer short" is sound advice about cost. In agent use, the cost lever is almost entirely on the input side, and the question becomes: ''what is in the context, and how long does it stay there?'' A log study by Arsentev (2026) analysed 722 coding-agent sessions run on two machines, covering 150,902 model calls and 34.6 billion tokens. Cost was modeled from the public price list rather than taken from invoices. More than 85% of the modeled cost came from context handling, meaning cache writes plus cache reads. The distribution was uneven: about 3% of sessions produced 80% of the cost, and sessions longer than 200 calls, only 8% of the total, produced more than 90% of the cost. This concentration shows why the input-side lever matters most in a small share of long-running sessions, rather than being spread evenly across all agent activity. === 1.1 Caching changes the arithmetic, but does not remove the effect === Because the same prefix is re-sent on every step, runtimes cache it. A cached prefix is charged at a reduced rate on subsequent reads, but writing it into the cache costs more than an ordinary input token. Per token, a cache ''write'' costs many times more than a cache ''read''; the exact ratio depends on the price list and on how long the cache is kept (a one-hour cache write is billed at twice the ordinary input rate). That single ratio is the whole story of this resource, because it creates a tension: * '''starting a new session often''' means the cache is written from scratch each time — many expensive writes; * '''never starting a new session''' means the context grows without limit, and every step re-reads a larger prefix — many cheap reads, but a great many of them. Two opposite failure modes, both expensive. Whenever a quantity is made worse by increasing ''and'' by decreasing a parameter, the curve between the extremes has a minimum somewhere in the middle. Finding it is an empirical question, not one to be settled by argument. == 2. Measuring a run correctly == Before any comparison can be made, a single run must be measured. Two mistakes are common enough that they deserve naming, and both were found the hard way in the experiment described in section 3. === 2.1 Deduplicate, but keep the largest record === A streaming runtime may emit several usage records for one model invocation: an early snapshot taken while the response is still being generated, and a final record once it completes. Both may carry the same message identifier. Counting both '''double counts''' the invocation. Counting only the first '''halves the output''', because the first record was written before generation finished. The correct procedure is therefore: # group usage records by their message or request identifier; # within each group, take the '''element-wise maximum''' across every usage field, rather than the first or the last record; # sum the resulting per-invocation records. A learner who takes only one lesson from this resource should take this one: a usage log is not a list of events to be added up, it is a set of observations of events, and observations of the same event must be reconciled before summation. === 2.2 Do not infer success from the exit status of a test runner === A comparison of cost is meaningless unless the work was actually done. It is tempting to record success by checking whether a test command exited zero, or by counting lines that look like test output. Test runners differ in what they print. A runner that emits a human-readable summary rather than a machine-readable protocol will defeat a line-counting heuristic silently — producing not an error but a wrong number, which is worse. Parse the runner's actual summary, and recompute pass and fail counts from the stored output rather than trusting a status recorded at the time. == 3. A controlled experiment == The following experiment is reported here as a worked example of the method above. Its data and analysis scripts are published; see [[#References|references]]. === 3.1 Design === A fixed set of '''12 self-contained programming tasks''' was defined against a growing synthetic library, each task requiring its tests to pass. The same 12 tasks were then executed under '''six conditions''', differing only in how many tasks were given to one session before a fresh session was started: : 1, 2, 3, 4, 6 and 12 tasks per session. One task per session is the behaviour of a user who clears context constantly; twelve tasks per session is the behaviour of a user who never clears it. Each condition was repeated '''6 times''', giving '''36 runs''' in total. Across all runs '''4,086 tests''' were executed with no failures, so the conditions can be compared on cost alone. === 3.2 Result === {| class="wikitable" |+ Cost and cache behaviour by session length (mean of 6 replicates) ! Tasks per session !! Sessions !! Model invocations !! Cache write (tokens) !! Cache read (tokens) !! Cost |- | 1 (clear always) || 12 || 94 || 252,533 || 3,622,795 || $2.17 |- | 2 || 6 || 84 || 169,670 || 3,578,718 || $1.85 |- | '''3''' || 4 || 72 || 131,100 || 3,160,517 || '''$1.62''' |- | 4 || 3 || 71 || 123,478 || 3,466,808 || $1.70 |- | 6 || 2 || 71 || 105,746 || 3,796,905 || $1.68 |- | 12 (never clear) || 1 || 63 || 87,846 || 4,451,367 || $1.78 |} Clearing after every task costs '''33.5% more''' than clearing every third task, for identical work with identical test outcomes. Never clearing costs 9.7% more in the point estimate, which is not significant after adjustment. (Costs corrected on 15 September 2026, see version 1.2 of the report: an earlier version priced cache reads and writes incorrectly and reported a 25.3% spread. Token counts and test results were not affected.) The mechanism is visible in the two cache columns and it is two-sided: * cache '''write''' falls monotonically as sessions get longer (252,533 → 87,846), because there are fewer fresh sessions to pay for; * context per invocation '''rises''' monotonically (approximately 38,000 → 71,000 tokens), because a long session accumulates; * their product, cache '''read''', is therefore U-shaped, and since one-hour cache writes are priced at 20 times reads, both ends of the range are expensive for different reasons. === 3.3 The bottom is a plateau, not a point === It would be easy, and wrong, to conclude "clear every third task". Exact two-sided permutation tests over all 924 partitions of six against six give: {| class="wikitable" |+ Comparisons against the cheapest condition ! Comparison !! ''p'' !! Interpretation |- | 3 vs 1 || 0.0022 || separated; this is the smallest value attainable at 6 versus 6 |- | 3 vs 12 || 0.046 || not significant after Holm adjustment over five comparisons (0.14) |- | 3 vs 2 || 0.0022 || separated |- | 3 vs 4 || 0.17 || '''not distinguishable''' |- | 3 vs 6 || 0.45 || '''not distinguishable''' |} So the defensible statement is not that three is best. It is that '''three, four and six are indistinguishable from one another, while clearing after every task is clearly worse'''; whether never clearing is also worse is not established at this sample size. The practical rule that follows is loose on purpose: work in batches of a few tasks per session, and do not adopt either habit that sits at an end of the range. === 3.4 What this experiment cannot establish === Honest limits matter more than headline numbers, and a learner should be able to state them: * one model, one task set, one growing repository; * six replicates per condition, which is enough to separate the extremes and not enough to separate neighbours; * the task set is synthetic and self-contained, so it understates the cost of exploration in an unfamiliar codebase; * pricing ratios are those of one provider at one time; the ''shape'' of the curve follows from write being dearer than read, but the position of its minimum does not transfer automatically. * a test suite that passes in every condition cannot tell the conditions apart on quality: all 4,086 tests passed with zero variance, so the comparison rests on cost alone. Why such a result set supports no conclusion about quality, and what a record should say about it, is argued in Arsentev (2026), ''A Suite That Does Not Discriminate''. Figures on this page follow the corrected version 1.2 of the report. Earlier versions, and an earlier three-replicate run, reported smaller spreads (25.3% and 32.9%) computed with incorrect prices; they should not be quoted. == 4. Exercises == # '''Arithmetic of accumulation.''' An agent run has 40 steps. The context is 5,000 tokens after step 1 and grows by 1,500 tokens per step. Compute the total input processed and compare it with a total output of 400 tokens per step. What fraction of the work is re-reading? # '''Reconciling a log.''' Take any usage log that contains duplicate records for one invocation. Compute the total three ways — summing everything, keeping the first of each duplicate, keeping the element-wise maximum — and explain which of the three errors would be invisible to a reader of the final number. # '''Break-even.''' Let ''w'' be the price of a cache write and ''r'' the price of a cache read per token, with ''w'' = ''a''·''r'' for some ratio ''a'' > 1. Take ''a'' from the current price list of the model you use. Write the cost of ''k'' tasks per session as a function of ''k'' under a linear model of context growth, and find the ''k'' that minimises it. Compare your answer with the table in section 3.2 and account for any difference. # '''Replication.''' Design the same comparison for a task set in a real repository rather than a synthetic one. State in advance which direction you expect the optimum to move, and why. # '''Critique.''' The experiment holds the task set fixed and varies only session length. Name one confound this design does ''not'' control for, and propose a modification that would. == 5. See also == * [[Large language models]] * [[Artificial intelligence]] * [[Machine learning]] == References == * Arsentev, E. (2026). ''The Context Economy of Agentic LLM Sessions: Where the Money Actually Goes''. Zenodo. Version 1.1 (corrected 15 September 2026). {{doi|10.5281/zenodo.22759216}} — log study of 722 agent sessions cited in section 1; costs are modeled from the public price list. * Arsentev, E. (2026). ''contextburn: a run-efficiency meter for language-model coding agents''. Zenodo. {{doi|10.5281/zenodo.22712985}} — open-source meter that reports, from local agent logs, the share of paid tokens that became output rather than context re-reading. * Arsentev, E. (2026). ''Clear Every Third Task: A Measured U-Curve in the Context Economy of Coding Agents''. Zenodo. Version 1.2 (corrected 15 September 2026). {{doi|10.5281/zenodo.22759217}} — report, dataset and analysis scripts for the experiment in section 3. Run-level data: {{doi|10.57967/hf/10366}}. * Arsentev, E. (2026). ''A Suite That Does Not Discriminate: The Case for Result-Set Fields in Agent Conformance Records''. Qeios. {{doi|10.32388/0BV3Z8}} — why a suite that passes everywhere cannot support a quality comparison. * Arsentev, E. (2026). ''Agent Run Metrics: A JSON Interchange Format for Resource Accounting of Language-Model Agent Runs''. Internet-Draft <code>draft-arsentev-agent-run-metrics-00</code>, IETF Datatracker — an interchange format for the per-run and per-step quantities used here, including a normative statement that one reported step corresponds to one completed model invocation. [[Category:Artificial intelligence]] [[Category:Machine learning]] [[Category:Computer science]] j8a7dsfpee6vric52uikr2srt1y4i5t 2834856 2834855 2026-09-28T14:45:37Z Evgenii Arsentev 3110906 Add cost-concentration chart (Commons, CC BY 4.0) 2834856 wikitext text/x-wiki {{Short description|Why agent runs cost what they do, and how to measure it}} This learning resource is a sub-page of [[Large language models]]. It is about a practical question that anyone running a coding agent meets within a week of starting: '''why does a session that does ordinary work cost what it costs, and what actually reduces that cost?''' The short answer is that the cost of an agent run is dominated not by the text the model writes, but by the text it reads again and again. This resource explains the mechanism, shows how to measure it without the two mistakes that make the measurement wrong, and works through a controlled experiment whose result contradicts the advice most commonly given. {{RightTOC}} == Learning objectives == After working through this resource a learner should be able to: * explain why the resource consumption of a multi-step agent run grows faster than the amount of text the agent produces; * distinguish '''cache write''' from '''cache read''' and say why their price difference determines the shape of the cost curve; * take a usage log from an agent runtime and compute the cost of a run correctly, avoiding double counting and truncated records; * design a controlled comparison between two ways of working with an agent, and state honestly what such a comparison can and cannot establish; * read a U-shaped cost curve and recognise when the bottom of it is a ''plateau'' rather than a point. == 1. Why repeated input dominates == An agent that works on a task over several steps does not send a fresh question to the model each time. It sends the accumulated conversation: the original instructions, every file it has read, every command it has run, and every answer it has produced so far. Then it appends the next step and sends the whole thing again. The consequence is worth stating as an inequality. If a run has ''n'' steps and the context after step ''i'' has length ''c_i'', the total input the model processes is : <math>\sum_{i=1}^{n} c_i</math> while the total output is only the sum of what was generated at each step, which in practical runs is one to two orders of magnitude smaller. A token admitted into the context on the first step of a sixty-step run is paid for sixty times. A token generated on the last step is paid for once. This is why intuitions carried over from single-turn use of a language model mislead. In single-turn use, "keep the answer short" is sound advice about cost. In agent use, the cost lever is almost entirely on the input side, and the question becomes: ''what is in the context, and how long does it stay there?'' [[File:Cost concentration in 722 coding-agent sessions (2026).svg|thumb|Cumulative share of modeled cost against share of sessions, 722 coding-agent sessions ranked by cost]] A log study by Arsentev (2026) analysed 722 coding-agent sessions run on two machines, covering 150,902 model calls and 34.6 billion tokens. Cost was modeled from the public price list rather than taken from invoices. More than 85% of the modeled cost came from context handling, meaning cache writes plus cache reads. The distribution was uneven: about 3% of sessions produced 80% of the cost, and sessions longer than 200 calls, only 8% of the total, produced more than 90% of the cost. This concentration shows why the input-side lever matters most in a small share of long-running sessions, rather than being spread evenly across all agent activity. === 1.1 Caching changes the arithmetic, but does not remove the effect === Because the same prefix is re-sent on every step, runtimes cache it. A cached prefix is charged at a reduced rate on subsequent reads, but writing it into the cache costs more than an ordinary input token. Per token, a cache ''write'' costs many times more than a cache ''read''; the exact ratio depends on the price list and on how long the cache is kept (a one-hour cache write is billed at twice the ordinary input rate). That single ratio is the whole story of this resource, because it creates a tension: * '''starting a new session often''' means the cache is written from scratch each time — many expensive writes; * '''never starting a new session''' means the context grows without limit, and every step re-reads a larger prefix — many cheap reads, but a great many of them. Two opposite failure modes, both expensive. Whenever a quantity is made worse by increasing ''and'' by decreasing a parameter, the curve between the extremes has a minimum somewhere in the middle. Finding it is an empirical question, not one to be settled by argument. == 2. Measuring a run correctly == Before any comparison can be made, a single run must be measured. Two mistakes are common enough that they deserve naming, and both were found the hard way in the experiment described in section 3. === 2.1 Deduplicate, but keep the largest record === A streaming runtime may emit several usage records for one model invocation: an early snapshot taken while the response is still being generated, and a final record once it completes. Both may carry the same message identifier. Counting both '''double counts''' the invocation. Counting only the first '''halves the output''', because the first record was written before generation finished. The correct procedure is therefore: # group usage records by their message or request identifier; # within each group, take the '''element-wise maximum''' across every usage field, rather than the first or the last record; # sum the resulting per-invocation records. A learner who takes only one lesson from this resource should take this one: a usage log is not a list of events to be added up, it is a set of observations of events, and observations of the same event must be reconciled before summation. === 2.2 Do not infer success from the exit status of a test runner === A comparison of cost is meaningless unless the work was actually done. It is tempting to record success by checking whether a test command exited zero, or by counting lines that look like test output. Test runners differ in what they print. A runner that emits a human-readable summary rather than a machine-readable protocol will defeat a line-counting heuristic silently — producing not an error but a wrong number, which is worse. Parse the runner's actual summary, and recompute pass and fail counts from the stored output rather than trusting a status recorded at the time. == 3. A controlled experiment == The following experiment is reported here as a worked example of the method above. Its data and analysis scripts are published; see [[#References|references]]. === 3.1 Design === A fixed set of '''12 self-contained programming tasks''' was defined against a growing synthetic library, each task requiring its tests to pass. The same 12 tasks were then executed under '''six conditions''', differing only in how many tasks were given to one session before a fresh session was started: : 1, 2, 3, 4, 6 and 12 tasks per session. One task per session is the behaviour of a user who clears context constantly; twelve tasks per session is the behaviour of a user who never clears it. Each condition was repeated '''6 times''', giving '''36 runs''' in total. Across all runs '''4,086 tests''' were executed with no failures, so the conditions can be compared on cost alone. === 3.2 Result === {| class="wikitable" |+ Cost and cache behaviour by session length (mean of 6 replicates) ! Tasks per session !! Sessions !! Model invocations !! Cache write (tokens) !! Cache read (tokens) !! Cost |- | 1 (clear always) || 12 || 94 || 252,533 || 3,622,795 || $2.17 |- | 2 || 6 || 84 || 169,670 || 3,578,718 || $1.85 |- | '''3''' || 4 || 72 || 131,100 || 3,160,517 || '''$1.62''' |- | 4 || 3 || 71 || 123,478 || 3,466,808 || $1.70 |- | 6 || 2 || 71 || 105,746 || 3,796,905 || $1.68 |- | 12 (never clear) || 1 || 63 || 87,846 || 4,451,367 || $1.78 |} Clearing after every task costs '''33.5% more''' than clearing every third task, for identical work with identical test outcomes. Never clearing costs 9.7% more in the point estimate, which is not significant after adjustment. (Costs corrected on 15 September 2026, see version 1.2 of the report: an earlier version priced cache reads and writes incorrectly and reported a 25.3% spread. Token counts and test results were not affected.) The mechanism is visible in the two cache columns and it is two-sided: * cache '''write''' falls monotonically as sessions get longer (252,533 → 87,846), because there are fewer fresh sessions to pay for; * context per invocation '''rises''' monotonically (approximately 38,000 → 71,000 tokens), because a long session accumulates; * their product, cache '''read''', is therefore U-shaped, and since one-hour cache writes are priced at 20 times reads, both ends of the range are expensive for different reasons. === 3.3 The bottom is a plateau, not a point === It would be easy, and wrong, to conclude "clear every third task". Exact two-sided permutation tests over all 924 partitions of six against six give: {| class="wikitable" |+ Comparisons against the cheapest condition ! Comparison !! ''p'' !! Interpretation |- | 3 vs 1 || 0.0022 || separated; this is the smallest value attainable at 6 versus 6 |- | 3 vs 12 || 0.046 || not significant after Holm adjustment over five comparisons (0.14) |- | 3 vs 2 || 0.0022 || separated |- | 3 vs 4 || 0.17 || '''not distinguishable''' |- | 3 vs 6 || 0.45 || '''not distinguishable''' |} So the defensible statement is not that three is best. It is that '''three, four and six are indistinguishable from one another, while clearing after every task is clearly worse'''; whether never clearing is also worse is not established at this sample size. The practical rule that follows is loose on purpose: work in batches of a few tasks per session, and do not adopt either habit that sits at an end of the range. === 3.4 What this experiment cannot establish === Honest limits matter more than headline numbers, and a learner should be able to state them: * one model, one task set, one growing repository; * six replicates per condition, which is enough to separate the extremes and not enough to separate neighbours; * the task set is synthetic and self-contained, so it understates the cost of exploration in an unfamiliar codebase; * pricing ratios are those of one provider at one time; the ''shape'' of the curve follows from write being dearer than read, but the position of its minimum does not transfer automatically. * a test suite that passes in every condition cannot tell the conditions apart on quality: all 4,086 tests passed with zero variance, so the comparison rests on cost alone. Why such a result set supports no conclusion about quality, and what a record should say about it, is argued in Arsentev (2026), ''A Suite That Does Not Discriminate''. Figures on this page follow the corrected version 1.2 of the report. Earlier versions, and an earlier three-replicate run, reported smaller spreads (25.3% and 32.9%) computed with incorrect prices; they should not be quoted. == 4. Exercises == # '''Arithmetic of accumulation.''' An agent run has 40 steps. The context is 5,000 tokens after step 1 and grows by 1,500 tokens per step. Compute the total input processed and compare it with a total output of 400 tokens per step. What fraction of the work is re-reading? # '''Reconciling a log.''' Take any usage log that contains duplicate records for one invocation. Compute the total three ways — summing everything, keeping the first of each duplicate, keeping the element-wise maximum — and explain which of the three errors would be invisible to a reader of the final number. # '''Break-even.''' Let ''w'' be the price of a cache write and ''r'' the price of a cache read per token, with ''w'' = ''a''·''r'' for some ratio ''a'' > 1. Take ''a'' from the current price list of the model you use. Write the cost of ''k'' tasks per session as a function of ''k'' under a linear model of context growth, and find the ''k'' that minimises it. Compare your answer with the table in section 3.2 and account for any difference. # '''Replication.''' Design the same comparison for a task set in a real repository rather than a synthetic one. State in advance which direction you expect the optimum to move, and why. # '''Critique.''' The experiment holds the task set fixed and varies only session length. Name one confound this design does ''not'' control for, and propose a modification that would. == 5. See also == * [[Large language models]] * [[Artificial intelligence]] * [[Machine learning]] == References == * Arsentev, E. (2026). ''The Context Economy of Agentic LLM Sessions: Where the Money Actually Goes''. Zenodo. Version 1.1 (corrected 15 September 2026). {{doi|10.5281/zenodo.22759216}} — log study of 722 agent sessions cited in section 1; costs are modeled from the public price list. * Arsentev, E. (2026). ''contextburn: a run-efficiency meter for language-model coding agents''. Zenodo. {{doi|10.5281/zenodo.22712985}} — open-source meter that reports, from local agent logs, the share of paid tokens that became output rather than context re-reading. * Arsentev, E. (2026). ''Clear Every Third Task: A Measured U-Curve in the Context Economy of Coding Agents''. Zenodo. Version 1.2 (corrected 15 September 2026). {{doi|10.5281/zenodo.22759217}} — report, dataset and analysis scripts for the experiment in section 3. Run-level data: {{doi|10.57967/hf/10366}}. * Arsentev, E. (2026). ''A Suite That Does Not Discriminate: The Case for Result-Set Fields in Agent Conformance Records''. Qeios. {{doi|10.32388/0BV3Z8}} — why a suite that passes everywhere cannot support a quality comparison. * Arsentev, E. (2026). ''Agent Run Metrics: A JSON Interchange Format for Resource Accounting of Language-Model Agent Runs''. Internet-Draft <code>draft-arsentev-agent-run-metrics-00</code>, IETF Datatracker — an interchange format for the per-run and per-step quantities used here, including a normative statement that one reported step corresponds to one completed model invocation. [[Category:Artificial intelligence]] [[Category:Machine learning]] [[Category:Computer science]] kbz9atbh0ga5ugo75ut0hp1ma5rze1f 2 332158 2834880 2834736 2026-09-28T19:33:17Z The Citer 3110681 2834880 wikitext text/x-wiki My template, text, and phrase storage! ==Templates== [[File:|thumb|300x300px|[Insert_text_here]]], [[File:|60px]] ==Text== – ==Phrases== <nowiki>''Incertae sedis''</nowiki> ==Purpose== It's a storage space so I don't have to copy text in source editor; I just can copy it in reading mode. guytcquzdd0u2noc1rttrbx9qk1qji1 0 332222 2834889 2834777 2026-09-28T19:55:18Z The Citer 3110681 Added Lycopodiophytes. 2834889 wikitext text/x-wiki [[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px|Different Plant species from different phyla.]] Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.<ref name=Jeffrey1967>{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}</ref><ref>{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}</ref> ==Information== Name Meaning: Twisted plant English Common Name: Streptophytes Major distinguishing characteristics: Approximate number of species described: >350,000 ==Classification== The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.<ref>{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}</ref>{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,<ref name=Jeffrey1967/><ref name=Adletal2012>{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}</ref> Streptophytina); others include more groups (e.g., Charophyta<!-- Karol et al. 2009--><ref name=Adletal2012/> Streptophyta,<ref name=Bremer1985>{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}</ref><!-- de Reviers 2002; --><ref name=Leliaert-2012>{{cite journal |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}</ref> Streptobionta<ref>{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}</ref>); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,<ref name="Cavalier-Smith1993">{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}</ref><ref name=Leliaert-2012/> Charophyceae{{citation needed|reason=Mattox & Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.<ref name=Liang2020>{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}</ref><ref name=Nedelcu2006>{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}</ref> ==Phylogeny== Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.<ref>{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology & Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}</ref><ref>{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}</ref><ref>{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}</ref> {{clade |label1=Streptophyta |1={{clade |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]` |2={{clade |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]] |2={{clade |1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]] |2={{clade |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]] |label2=Phragmoplastophyta |2={{clade |1=Charophyceae (Stoneworts & musk grasses) [[File:CharaFragilis.jpg|60px]] |2={{clade |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]] |2={{clade |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]] |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]] }} }} }} }} }} }} }}}} ==Diversity== There are many species of Streptophytes. These are just the phyla: ===Anthocerotophytes=== [[File:Dendroceros.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]<br> Name Meaning: Anthoceros-like plant English Common Name: Hornworts Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system Approximate number of species described: 100-300 Classes: Anthocerotopsida, Leiosporocerotopsida ===Bryophytes=== [[File:Tionesta-ac-moss2.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Bryophyta]]<br> Name Meaning: Bryum-like plant, moss plant English Common Name: Moss Major distinguishing characteristics: Persistent branched sporophytes, no vascular system Approximate number of species described: 12,000 Classes: Andreaeobryopsida, Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida ===Cycadophytes=== [[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Cycadophyta]]<br> Name Meaning: Cycas-like plant, palm-like plant English Common Name: Cycads Major distinguishing characteristics: Seeds, crown of compound leaves Approximate number of species described: 100 - 200 Classes: Cycadopsida ===Ginkgophytes=== [[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]<br> Name Meaning: Ginkgo-like plant English Common Name: Ginkgo, maidenhair tree Major distinguishing characteristics: Seeds not protected by fruit Approximate number of species described: 1 living, about 50 extinct Classes: Ginkgoopsida ===Gnetophytes=== [[File:Welwitschia at Ugab River basin.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Gnetophyta]]<br> Name Meaning: Gnetum-like plant English Common Name: Gnetophytes Major distinguishing characteristics: Seeds and woody vascular system with vessels. Approximate number of species described: 70 Classes: Gnetopsida—Actually, it's a subclass: Gnetidae ===Lycopodiophytes=== [[File:Clubmoss - Flickr - pellaea (1).jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Lycopodiophyta (Lycophyta)]]<br> Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living Classes: Lycopodiopsida, †Zosterophyllopsida ====Clades ''Incertae sedis''==== †Drepanophycales – †''Nothia'' ==References== [[Category:Plants]] [[Category:Taxonomy]] [[Category:Botany]] [[Category:Biology]] 2c9dtz7087awsh5c53b5f8qcgsn5po8 2834924 2834889 2026-09-28T22:35:02Z The Citer 3110681 Angiospermae 2834924 wikitext text/x-wiki [[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px|Different Plant species from different phyla.]] Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.<ref name=Jeffrey1967>{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}</ref><ref>{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}</ref> ==Information== Name Meaning: Twisted plant English Common Name: Streptophytes Major distinguishing characteristics: Approximate number of species described: >350,000 ==Classification== The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.<ref>{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}</ref>{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,<ref name=Jeffrey1967/><ref name=Adletal2012>{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}</ref> Streptophytina); others include more groups (e.g., Charophyta<!-- Karol et al. 2009--><ref name=Adletal2012/> Streptophyta,<ref name=Bremer1985>{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}</ref><!-- de Reviers 2002; --><ref name=Leliaert-2012>{{cite journal |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}</ref> Streptobionta<ref>{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}</ref>); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,<ref name="Cavalier-Smith1993">{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}</ref><ref name=Leliaert-2012/> Charophyceae{{citation needed|reason=Mattox & Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.<ref name=Liang2020>{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}</ref><ref name=Nedelcu2006>{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}</ref> ==Phylogeny== Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.<ref>{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology & Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}</ref><ref>{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}</ref><ref>{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}</ref> {{clade |label1=Streptophyta |1={{clade |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]` |2={{clade |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]] |2={{clade |1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]] |2={{clade |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]] |label2=Phragmoplastophyta |2={{clade |1=Charophyceae (Stoneworts & musk grasses) [[File:CharaFragilis.jpg|60px]] |2={{clade |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]] |2={{clade |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]] |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]] }} }} }} }} }} }} }}}} ==Diversity== There are many species of Streptophytes. These are just the phyla: ===Anthocerotophytes=== [[File:Dendroceros.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]<br> Name Meaning: Anthoceros-like plant English Common Name: Hornworts Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system Approximate number of species described: 100-300 Classes: Anthocerotopsida, Leiosporocerotopsida ===Bryophytes=== [[File:Tionesta-ac-moss2.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Bryophyta]]<br> Name Meaning: Bryum-like plant, moss plant English Common Name: Moss Major distinguishing characteristics: Persistent branched sporophytes, no vascular system Approximate number of species described: 12,000 Classes: Andreaeobryopsida, Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida ===Cycadophytes=== [[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Cycadophyta]]<br> Name Meaning: Cycas-like plant, palm-like plant English Common Name: Cycads Major distinguishing characteristics: Seeds, crown of compound leaves Approximate number of species described: 100 - 200 Classes: Cycadopsida ===Ginkgophytes=== [[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]<br> Name Meaning: Ginkgo-like plant English Common Name: Ginkgo, maidenhair tree Major distinguishing characteristics: Seeds not protected by fruit Approximate number of species described: 1 living, about 50 extinct Classes: Ginkgoopsida ===Gnetophytes=== [[File:Welwitschia at Ugab River basin.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Gnetophyta]]<br> Name Meaning: Gnetum-like plant English Common Name: Gnetophytes Major distinguishing characteristics: Seeds and woody vascular system with vessels. Approximate number of species described: 70 Classes: Gnetopsida—Actually, it's a subclass: Gnetidae ===Lycopodiophytes=== [[File:Clubmoss - Flickr - pellaea (1).jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Lycopodiophyta (Lycophyta)]]<br> Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living Classes: Lycopodiopsida, †Zosterophyllopsida ====Clades ''Incertae sedis''==== †Drepanophycales – †''Nothia'' ===Magnoliophytes=== [[File:Acer pseudoplatanus Chaltenbrunnen.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Magnoliophyta]]<br> Name Meaning: Magnolia-like plant English Common Name: Flowering plants, angiosperms Major distinguishing characteristics: Flowers and fruit, vascular system with vessels Approximate number of species described: 300,000 Classes: Magnoliopsida ==References== [[Category:Plants]] [[Category:Taxonomy]] [[Category:Botany]] [[Category:Biology]] tal37maase3zz5n7b2utxq5cunu6b4b 2835022 2834924 2026-09-29T10:36:02Z The Citer 3110681 Liverworts 2835022 wikitext text/x-wiki [[File:Diversity of plants (Streptophyta) version 2.png|thumb|500x500px|Different Plant species from different phyla.]] Streptophyta (/strɛpˈtɒfɪtə, ˈstrɛptoʊfaɪtə/), informally the streptophytes (/ˈstrɛptəfaɪts/, from the Greek strepto 'twisted', for the morphology of the sperm of some members), is a clade of plants. The composition of the clade varies considerably between authors, but the definition employed here includes land plants and all green algae except the Chlorophyta and the more basal Prasinodermophyta.<ref name=Jeffrey1967>{{cite journal | author = Jeffrey C | year = 1967 | title = The origin and differentiation of the Archegoniate land plants: A second contribution | journal = Kew Bull. | volume = 21 | issue = 2| pages = 335–349 | doi=10.2307/4108533| jstor = 4108533 | bibcode = 1967KewBu..21..335J }}</ref><ref>{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}</ref> ==Information== Name Meaning: Twisted plant English Common Name: Streptophytes Major distinguishing characteristics: Approximate number of species described: >350,000 ==Classification== The composition of Streptophyta and similar groups (Streptophytina, [[Plant_Divisions_(Phyla)/Charophyta|Charophyta]]) varies in each classification.<ref>{{Cite web |title=Die Schwestergruppe der Landpflanzen |trans-title=The sister group of land plants |url=https://www.protisten.de/german/docs/Phylogeny_in_Streptophyta.pdf |access-date=7 November 2023 |website=www.protisten.de |language=de}}</ref>{{citation needed|reason=cited source is in German. An English-language source is required|date=September 2026}} Some authors include only the Charales and Embryophyta (e.g., Streptophyta,<ref name=Jeffrey1967/><ref name=Adletal2012>{{cite journal |author=Adl, S.M.; Simpson, A.G.B.; Lane, C.E.; Lukeš, J.; Bass, D.; Bowser, S.S.; Brown, M.W.; Burki, F.; Dunthorn, M mentioned as source without citation.; Hampl, V.; Heiss, A.; Hoppenrath, M.; Lara, E.; le Gall, L.; Lynn, D.H.; McManus, H.; Mitchell, E.A.D.; Mozley-Stanridge, S.E.; Parfrey, L.W.; Pawlowski, J.; Rueckert, S.; Shadwick, L.; Schoch, C.L.; Smirnov, A.; Spiegel, F.W. |date=2012 |title=The revised classification of Eukaryotes |journal=Journal of Eukaryotic Microbiology |volume=59 |issue=5 |pages=429–514 |doi=10.1111/j.1550-7408.2012.00644.x| pmc=3483872|pmid=23020233 }}</ref> Streptophytina); others include more groups (e.g., Charophyta<!-- Karol et al. 2009--><ref name=Adletal2012/> Streptophyta,<ref name=Bremer1985>{{cite journal |author=Bremer, K. |date=1985 |title=Summary of green plant phylogeny and classification |journal=Cladistics |volume=1 |issue=4 |pages=369–385 |doi=10.1111/j.1096-0031.1985.tb00434.x |pmid=34965683 }}</ref><!-- de Reviers 2002; --><ref name=Leliaert-2012>{{cite journal |last1=Leliaert |first1=Frederik |last2=Smith |first2=David R. |last3=Moreau |first3=Hervé |last4=Herron |first4=Matthew D. |last5=Verbruggen |first5=Heroen |last6=Delwiche |first6=Charles F. |last7=De Clerck |first7=Olivier |title=Phylogeny and Molecular Evolution of the Green Algae |date=2012 |journal=Critical Reviews in Plant Sciences |volume=31 |issue=1 |pages=1–46 |bibcode=2012CRvPS..31....1L |doi=10.1080/07352689.2011.615705 |url=http://www.vliz.be/imisdocs/publications/248853.pdf }}</ref> Streptobionta<ref>{{cite book|last1=Kenrick|first1=Paul|last2=Crane|first2=Peter|title=The Origin and Early Diversification of Land Plants: A Cladistic Study|date=1997|publisher=[[Smithsonian Institution Press]]|location=Washington, D.C. |isbn=9781560987291}}</ref>); some authors use this broader definition, but exclude the Embryophyta (e.g., Charophyta,<ref name="Cavalier-Smith1993">{{cite book|author=Cavalier-Smith, T. |date=1993| chapter=The origin, losses and gains of chloroplasts. |title=Origins of plastids |pages=291–348 |publisher=Springer US}}</ref><ref name=Leliaert-2012/> Charophyceae{{citation needed|reason=Mattox & Stewart, 1984 mentioned as source without citation|date=September 2026}}, Streptophycophytes{{citation needed|reason=de Reviers, 2002 meantion as source without citation|date=April 2024}}). The clade Streptophyta includes both unicellular and multicellular organisms. Streptophyta contains the freshwater charophyte green algae and all land plants that reproduce sexually by conjugation. ''Mesostigma viride'', a unicellular green flagellate alga may be a basal Streptophyte.<ref name=Liang2020>{{cite journal |last1=Liang |first1=Z |last2=Geng |first2=Y |last3=Ji |first3=C |last4=Du |first4=H |last5=Wong |first5=CE |last6=Zhang |first6=Q |last7=Yu |first7=H |date=2020 |title=''Mesostigma viride'' genome and transcriptome provide insights into the origin and evolution of Streptophyta |journal=Advanced Science |volume=7 |issue=1 |doi=10.1002/advs.2019018501901850 |doi-broken-date=11 August 2026 |doi-access=free }}</ref><ref name=Nedelcu2006>{{cite journal |last1=Nedelcu |first1=AM |last2=Borza |first2=T |last3=Lee |first3=RW |date=2006 |title=A land plant–specific multigene family in the unicellular ''Mesostigma'' argues for its close relationship to Streptophyta |journal=Molecular Biology and Evolution |volume=23 |issue=5 |pages=1011–1015 |doi=10.1093/molbev/msj108 |pmid=16476689 }}</ref> ==Phylogeny== Below is a reconstruction of '''Streptophyta''' relationships, based on genomic data.<ref>{{Cite journal|author1=Linzhou Li|author2=Sibo Wang|author3=Hongli Wang|author4=Sunil Kumar Sahu|author5=Birger Marin|author6=Haoyuan Li|author7=Yan Xu|author8=Hongping Liang|author9=Zhen Li|author10=Shifeng Chen|author11=Tanja Reder|date=22 June 2020|title=The genome of Prasinoderma coloniale unveils the existence of a third phylum within green plants|journal=Nature Ecology & Evolution|volume=4|issue=9|pages=1220–1231|doi=10.1038/s41559-020-1221-7|pmc=7455551|pmid=32572216|doi-access=free|author22=Yves Van de Peer|author23=Michael Melkonian|author24=Huan Liu|author21=Xin Liu|author20=Xun Xu|author19=Gane Ka-Shu Wong|author17=Huanming Yang|author16=Hongli Du|author15=Barbara Melkonian|author14=Morten Petersen|author13=Sebastian Wittek|author12=Zehra Çebi|author18=Jian Wang|bibcode=2020NatEE...4.1220L }}</ref><ref>{{Cite journal|last1=Puttick|first1=Mark N.|last2=Morris|first2=Jennifer L.|last3=Williams|first3=Tom A.|last4=Cox|first4=Cymon J.|last5=Edwards|first5=Dianne|last6=Kenrick|first6=Paul|last7=Pressel|first7=Silvia|last8=Wellman|first8=Charles H.|last9=Schneider|first9=Harald|date=2018|title=The Interrelationships of Land Plants and the Nature of the Ancestral Embryophyte|journal=Current Biology|volume=28|issue=5|pages=733–745.e2|doi=10.1016/j.cub.2018.01.063|pmid=29456145|doi-access=free|bibcode=2018CBio...28E.733P |hdl=1983/ad32d4da-6cb3-4ed6-add2-2415f81b46da|hdl-access=free}}</ref><ref>{{Cite journal|last1=Sánchez-Baracaldo|first1=Patricia|last2=Raven|first2=John A.|last3=Pisani|first3=Davide|last4=Knoll|first4=Andrew H.|date=2017-09-12|title=Early photosynthetic eukaryotes inhabited low-salinity habitats|journal=Proceedings of the National Academy of Sciences|volume=114|issue=37|pages=E7737–E7745|doi=10.1073/pnas.1620089114|url=https://research-information.bristol.ac.uk/ws/files/132985852/pnas.1620089114.sapp.pdf|pmid=28808007|pmc=5603991|bibcode=2017PNAS..114E7737S |doi-access=free}}</ref> {{clade |label1=Streptophyta |1={{clade |1=Mesostigmatophyceae [[File:Mesostigma viride 534857918.jpg|60px]]` |2={{clade |1=Chlorokybophyceae [[File:Chlorokybus atmophyticus.jpg|60px]] |2={{clade |1=''Streptofilum'' [[File:Streptofilum capillatum.pdf|60px]] |2={{clade |1=Klebsormidiophyceae [[File:Klebsormidium bilatum Belgium (14759117646).jpg|60px]] |label2=Phragmoplastophyta |2={{clade |1=Charophyceae (Stoneworts & musk grasses) [[File:CharaFragilis.jpg|60px]] |2={{clade |1=Zygnematophyceae (desmids, water silk etc.) [[File:The freshwater alga Spirogyra.jpg|60px]] |2={{clade |1=Coleochaetophyceae [[File:Spirogyra-bgiu.jpg|60px]] |2=Embryophytes [[File:GINKGOBAUM-2.jpg|60px]] }} }} }} }} }} }} }}}} ==Diversity== There are many species of Streptophytes. These are just the phyla: ===Anthocerotophytes=== [[File:Dendroceros.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Anthocerotophyta]]<br> Name Meaning: Anthoceros-like plant English Common Name: Hornworts Major distinguishing characteristics: Horn-shaped sporophytes, no vascular system Approximate number of species described: 100-300 Classes: Anthocerotopsida, Leiosporocerotopsida ===Bryophytes=== [[File:Tionesta-ac-moss2.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Bryophyta]]<br> Name Meaning: Bryum-like plant, moss plant English Common Name: Moss Major distinguishing characteristics: Persistent branched sporophytes, no vascular system Approximate number of species described: 12,000 Classes: Andreaeobryopsida, Andreaeopsida, Bryopsida, Oedipodiopsida, Sphagnopsida, Takakiopsida, Tetraphidopsida ===Cycadophytes=== [[File:Starr 060905-8736 Zamia furfuracea.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Cycadophyta]]<br> Name Meaning: Cycas-like plant, palm-like plant English Common Name: Cycads Major distinguishing characteristics: Seeds, crown of compound leaves Approximate number of species described: 100 - 200 Classes: Cycadopsida ===Ginkgophytes=== [[File:Ginkgo biloba in Lucenec2.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Ginkgophyta]]<br> Name Meaning: Ginkgo-like plant English Common Name: Ginkgo, maidenhair tree Major distinguishing characteristics: Seeds not protected by fruit Approximate number of species described: 1 living, about 50 extinct Classes: Ginkgoopsida ===Gnetophytes=== [[File:Welwitschia at Ugab River basin.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Gnetophyta]]<br> Name Meaning: Gnetum-like plant English Common Name: Gnetophytes Major distinguishing characteristics: Seeds and woody vascular system with vessels. Approximate number of species described: 70 Classes: Gnetopsida—Actually, it's a subclass: Gnetidae ===Lycopodiophytes=== [[File:Clubmoss - Flickr - pellaea (1).jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Lycopodiophyta (Lycophyta)]]<br> Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living Classes: Lycopodiopsida, †Zosterophyllopsida ====Clades ''Incertae sedis''==== †Drepanophycales – †''Nothia'' ===Magnoliophytes=== [[File:Acer pseudoplatanus Chaltenbrunnen.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Magnoliophyta]]<br> Name Meaning: Magnolia-like plant English Common Name: Flowering plants, angiosperms Major distinguishing characteristics: Flowers and fruit, vascular system with vessels Approximate number of species described: 300,000 Classes: Magnoliopsida ===Marchantiophytes=== [[File:Lunularia cruciata.jpg|thumb|200x200px]] Main article: [[Plant Divisions (Phyla)/Marchantiophyta]]<br> Name Meaning: Marchantia-like plant, liver plant English Common Name: Liverworts Major distinguishing characteristics: Ephemeral unbranched sporophytes, no vascular system Approximate number of species described: 9,000 Classes: Haplomitriopsida, Jungermanniopsida, Marchantiopsida ==References== [[Category:Plants]] [[Category:Taxonomy]] [[Category:Botany]] [[Category:Biology]] tivcf92zd4nbej2n1407ayubrfhjx7z 0 332225 2834849 2834701 2026-09-28T12:30:01Z 202 Xup iartkab lakarg 3111500 /* Example: Author's Lost Zodiac natal chart */ 2834849 wikitext text/x-wiki {{Infobox | title = The Lost Zodiac | image = | caption = | label1 = Author | data1 = Tahmid Arian Borno | label2 = Year | data2 = 2026 | label3 = Status | data3 = Proposed hypothesis; under development | label4 = Subject | data4 = Astrology / stellar symbolism }} == The Lost Zodiac == '''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026. It is an experimental astrological framework based on selected stars and constellational figures, their associated date ranges, and a proposed method for converting conventional natal-chart positions into Lost Zodiac positions. <div style="clear: both; width: auto; margin: 1em 0;"> <!-- Research Project Box --> <div style="border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;"> '''This is a research project at Wikiversity.''' </div> <!-- Hypothesis Warning Box --> <div style="border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;"> '''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.''' </div> </div> == Background == The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year. The Lost Zodiac system used as the basis for this project was created by astrologer Catherine Tennant and features 22 "lost" star signs based on major constellations located north and south of the main zodiac belt.<ref name="Interactive Stars">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac_sign.php |title=The Lost Zodiac |publisher=InteractiveStars.com |access-date=2026-09-23 }}</ref> The period boundaries, associated figures, and principal stars used in this project are derived directly from that source; the organization of the periods within this article and the natal-chart conversion methodology described below remain original to the author. The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction. == The Lost Zodiac Signs == {| class="wikitable sortable" ! Sign ! Period ! Associated star(s) ! Sources |- | The Lyre of Orpheus | December 29 – January 13 | Vega | <ref name="The Lyre of Orpheus">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&date=Dec-29th@Jan-13th |title=The Lyre of Orpheus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Eagle | January 14 – January 28 | Altair | <ref name="The Eagle">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf |title=The Eagle |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Dolphin | January 29 – February 8 | Rotanev and Sualocin | <ref name="The Dolphin">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&date=Jan-29th@Feb-8th |title=The Dolphin |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Swan | February 9 – February 28/29 | Sadir | <ref name="The Swan">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf |title=The Swan |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The River of Night | March 1 – March 12 | Achernar | <ref name="The River of Night">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf |title=The River of Night |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Pegasus | March 13 – April 1 | Markab, Scheat and Algenib | <ref name="Pegasus">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf |title=Pegasus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Andromeda | April 2 – April 9 | Alpheratz | <ref name="Andromeda">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf |title=Andromeda |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The River of Night | April 10 – April 18 | Acamar | <ref name="The River of Night"/> |- | Andromeda | April 19 – May 8 | Mirach and Alamach | <ref name="Andromeda"/> |- | The River of Night | May 9 – May 15 | Rana and Zanrak | <ref name="The River of Night"/> |- | Perseus | May 16 – May 31 | Algol and Mirfak | <ref name="Perseus">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&date=May16th@31st |title=Perseus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Orion | June 1 – June 7 | Rigel | <ref name="Orion">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf |title=Orion |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Charioteer | June 8 – June 16 | Capella | <ref name="The Charioteer">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf |title=The Charioteer |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Orion | June 17 – June 27 | Betelgeuse | <ref name="Orion"/> |- | The Dogs | June 28 – July 7 | Sirius | <ref name="The Dogs">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf |title=The Dogs |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Ship of the Argonauts | July 8 – July 17 | Canopus | <ref name="Ship of the Argonauts">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf |title=The Ship of the Argonauts |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Dogs | July 18 – July 25 | Procyon | <ref name="The Dogs"/> |- | The Dragon | July 26 – August 7 | Gianfar | <ref name="The Dragon">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf |title=The Dragon |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Great Bear | August 8 – August 15 | Dubhe and Merak | <ref name="The Great Bear">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&date=Aug-8th@15th |title=The Great Bear |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Sea Serpent | August 16 – August 23 | Alphard | <ref name="The Sea Serpent">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf |title=The Sea Serpent |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Great Bear | August 24 – September 10 | Phekda, Megrez, Alioth and Mizar | <ref name="The Great Bear"/> |- | The Cup | September 11 – September 21 | Alkes | <ref name="The Cup">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf |title=The Cup |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Ship of the Argonauts | September 22 – September 28 | Markeb | <ref name="Ship of the Argonauts"/> |- | The Raven | September 29 – October 11 | Minkar and Algorab | <ref name="The Raven">{{cite web |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72 |title=The Raven |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Bear Keeper | October 12 – October 26 | Arcturus and Izar | <ref name="The Bear Keeper">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf |title=The Bear Keeper |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Crown of the North Wind | October 27 – November 10 | Alphecca | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf |title=The Lost Zodiac |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Serpent | November 11 – November 19 | Unuk Elhaia | <ref>{{cite web |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ |title=The Serpent |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Wise Centaur | November 20 – December 5 | Toliman (Alpha Centauri) | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf |title=The Wise Centaur |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Ophiuchus | December 6 – December 16 | Han, Sabik and Ras Alhague | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf |title=Ophiuchus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Dragon | December 17 – December 23 | Grumium and Etanin | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf |title=The Dragon |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Serpent | December 24 – December 28 | Alya | <ref>{{cite web |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ |title=The Serpent |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |} == Natal-chart conversion methodology == The Lost Zodiac proposes a proportional conversion between a conventional zodiacal position and the corresponding Lost Zodiac period. A placement is first treated as a position within its conventional zodiac sign. That position is converted into a corresponding calendar date and time. The resulting date and time are then located within the relevant Lost Zodiac period. Each listed period includes the full span of every calendar date in its range. The starting date begins at 12:00 a.m., and the ending date concludes at 11:59 p.m. For example, a period ending on February 28 includes the entire day of February 28. In leap years, the Swan period ends at 11:59 p.m. on February 29. For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and the end of the period as 29°59′. The beginning boundary is midnight at the start of the first listed date, while the ending boundary is 11:59 p.m. on the last listed date. The proposed formula is: <math> L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59' </math> where: * <math>L</math> = resulting Lost Zodiac degree * <math>t</math> = the corresponding date and time of the natal placement * <math>t_0</math> = beginning of the Lost Zodiac period * <math>t_1</math> = end boundary of the Lost Zodiac period === Example === A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009 at approximately 6:25 AM in the example used by the author. The Swan period runs from February 9 through the last day of February, ending on February 28 in a common year or February 29 in a leap year. The position of February 20 at 6:25 AM is approximately 56.34% through the Swan period. Therefore: <math> 29^\circ 59'\times0.5634\approx16^\circ 53' </math> The resulting proposed Lost Zodiac position is therefore: '''Swan 16°53′''' == Reference natal chart == The following conventional natal-chart placements are used as the reference positions for the Lost Zodiac conversion. {| class="wikitable" ! Natal point ! Conventional natal position ! House |- | Sun | Pisces 1°29′ | 1st House |- | Ascendant | Aquarius 29°24′ | — |- | Moon | Capricorn 5°24′ | 11th House |- | Mercury | Aquarius 6°07′ | 12th House |- | Venus | Aries 11°30′ | 2nd House |- | Mars | Aquarius 11°55′ | 12th House |- | Jupiter | Aquarius 10°38′ | 12th House |- | Saturn | Virgo 19°41′ Rx | 7th House |- | Uranus | Pisces 21°25′ | 1st House |- | Neptune | Aquarius 24°13′ | 12th House |- | Pluto | Capricorn 2°47′ | 10th House |- | North Node | Aquarius 8°18′ Rx | 12th House |- | Lilith | Capricorn 5°16′ | 11th House |- | Chiron | Aquarius 21°52′ | 12th House |- | Fortune | Aries 25°29′ | 2nd House |- | Vertex | Virgo 17°15′ | 7th House |- | MC | Sagittarius 8°28′ | — |} == Example: Author's Lost Zodiac natal chart == Using the conversion methodology described above, the author's proposed Lost Zodiac natal placements are: {| class="wikitable" ! Natal point ! Lost Zodiac placement |- | Sun | Swan 16°53′ |- | Ascendant | Swan 14°06′ |- | Moon | Serpent 20°23′ |- | Mercury | Eagle 24°13′ |- | Venus | Pegasus 29°14′ |- | Mars | Dolphin 7°57′ |- | Jupiter | Dolphin 4°27′ |- | Saturn | Cup 1°52′ |- | Uranus | River of Night 28°32′ |- | Neptune | Swan 6°19′ |- | Pluto | Serpent 4°42′ |- | North Node | Eagle 28°35′ |- | Lilith | Serpent 19°35′ |- | Chiron | Swan 2°48′ |- | Fortune | River of Night 18°16′ |- | Vertex | Great Bear 27°04′ |- | MC | Wise Centaur 19°37′ |} l9cttyfmoj3sdsepbqxxnmola2c4375 2834850 2834849 2026-09-28T12:30:49Z 202 Xup iartkab lakarg 3111500 /* Natal chart conversion methodology */ 2834850 wikitext text/x-wiki {{Infobox | title = The Lost Zodiac | image = | caption = | label1 = Author | data1 = Tahmid Arian Borno | label2 = Year | data2 = 2026 | label3 = Status | data3 = Proposed hypothesis; under development | label4 = Subject | data4 = Astrology / stellar symbolism }} == The Lost Zodiac == '''The Lost Zodiac''' is a proposed hypothesis developed by [[Tahmid Arian Borno]] in 2026. It is an experimental astrological framework based on selected stars and constellational figures, their associated date ranges, and a proposed method for converting conventional natal-chart positions into Lost Zodiac positions. <div style="clear: both; width: auto; margin: 1em 0;"> <!-- Research Project Box --> <div style="border-left: 10px solid #0066cc; background-color: #f0f8ff; padding: 0.5em 1em; margin-bottom: 0.5em; box-sizing: border-box; width: 100%;"> '''This is a research project at Wikiversity.''' </div> <!-- Hypothesis Warning Box --> <div style="border-left: 10px solid #f28500; background-color: #fef6e7; padding: 0.5em 1em; box-sizing: border-box; width: 100%;"> '''This article describes a personal hypothesis that is currently under development. It is not presented as an established astronomical or astrological system. The astronomical information underlying individual stars and constellations is supported by external sources; the organization of the periods and the natal-chart conversion methodology are proposed by the author.''' </div> </div> == Background == The Lost Zodiac is based on a set of zodiacal periods associated with prominent stars and constellational figures. The source system on which the reconstruction is based contains 31 periods representing 22 distinct constellational figures. Several figures recur in non-adjacent periods because different stars associated with the same figure are assigned to different parts of the year. The Lost Zodiac system used as the basis for this project was created by astrologer Catherine Tennant and features 22 "lost" star signs based on major constellations located north and south of the main zodiac belt.<ref name="Interactive Stars">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac_sign.php |title=The Lost Zodiac |publisher=InteractiveStars.com |access-date=2026-09-23 }}</ref> The period boundaries, associated figures, and principal stars used in this project are derived directly from that source; the organization of the periods within this article and the natal-chart conversion methodology described below remain original to the author. The present project does not claim that the organization of these periods or their relationship to the modern natal zodiac has been historically established. Instead, the source material is used as the astronomical and symbolic foundation for a proposed reconstruction. == The Lost Zodiac Signs == {| class="wikitable sortable" ! Sign ! Period ! Associated star(s) ! Sources |- | The Lyre of Orpheus | December 29 – January 13 | Vega | <ref name="The Lyre of Orpheus">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=orpheus&date=Dec-29th@Jan-13th |title=The Lyre of Orpheus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Eagle | January 14 – January 28 | Altair | <ref name="The Eagle">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Eagle.pdf |title=The Eagle |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Dolphin | January 29 – February 8 | Rotanev and Sualocin | <ref name="The Dolphin">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=dolphin&date=Jan-29th@Feb-8th |title=The Dolphin |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Swan | February 9 – February 28/29 | Sadir | <ref name="The Swan">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Swan.pdf |title=The Swan |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The River of Night | March 1 – March 12 | Achernar | <ref name="The River of Night">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_River_of_Night.pdf |title=The River of Night |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Pegasus | March 13 – April 1 | Markab, Scheat and Algenib | <ref name="Pegasus">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Pegasus.pdf |title=Pegasus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Andromeda | April 2 – April 9 | Alpheratz | <ref name="Andromeda">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Andromeda.pdf |title=Andromeda |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The River of Night | April 10 – April 18 | Acamar | <ref name="The River of Night"/> |- | Andromeda | April 19 – May 8 | Mirach and Alamach | <ref name="Andromeda"/> |- | The River of Night | May 9 – May 15 | Rana and Zanrak | <ref name="The River of Night"/> |- | Perseus | May 16 – May 31 | Algol and Mirfak | <ref name="Perseus">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=perseus&date=May16th@31st |title=Perseus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Orion | June 1 – June 7 | Rigel | <ref name="Orion">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Orion.pdf |title=Orion |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Charioteer | June 8 – June 16 | Capella | <ref name="The Charioteer">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Charioteer.pdf |title=The Charioteer |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Orion | June 17 – June 27 | Betelgeuse | <ref name="Orion"/> |- | The Dogs | June 28 – July 7 | Sirius | <ref name="The Dogs">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Dogs.pdf |title=The Dogs |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Ship of the Argonauts | July 8 – July 17 | Canopus | <ref name="Ship of the Argonauts">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Ship_of_the_Argonauts.pdf |title=The Ship of the Argonauts |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Dogs | July 18 – July 25 | Procyon | <ref name="The Dogs"/> |- | The Dragon | July 26 – August 7 | Gianfar | <ref name="The Dragon">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf |title=The Dragon |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Great Bear | August 8 – August 15 | Dubhe and Merak | <ref name="The Great Bear">{{cite web |url=https://www.interactivestars.com/beyond_zodiac/lost_zodiac.php?sign=great_bear&date=Aug-8th@15th |title=The Great Bear |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Sea Serpent | August 16 – August 23 | Alphard | <ref name="The Sea Serpent">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Sea_Serpent.pdf |title=The Sea Serpent |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Great Bear | August 24 – September 10 | Phekda, Megrez, Alioth and Mizar | <ref name="The Great Bear"/> |- | The Cup | September 11 – September 21 | Alkes | <ref name="The Cup">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Cup.pdf |title=The Cup |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Ship of the Argonauts | September 22 – September 28 | Markeb | <ref name="Ship of the Argonauts"/> |- | The Raven | September 29 – October 11 | Minkar and Algorab | <ref name="The Raven">{{cite web |url=https://jumpshare.com/share/plV6yw6ulAAMDx81uL72 |title=The Raven |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Bear Keeper | October 12 – October 26 | Arcturus and Izar | <ref name="The Bear Keeper">{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Bear_Keeper.pdf |title=The Bear Keeper |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Crown of the North Wind | October 27 – November 10 | Alphecca | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Crown_of_the_North_Wind.pdf |title=The Lost Zodiac |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Serpent | November 11 – November 19 | Unuk Elhaia | <ref>{{cite web |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ |title=The Serpent |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Wise Centaur | November 20 – December 5 | Toliman (Alpha Centauri) | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Wise_Centaur.pdf |title=The Wise Centaur |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | Ophiuchus | December 6 – December 16 | Han, Sabik and Ras Alhague | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/Ophiuchus.pdf |title=Ophiuchus |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Dragon | December 17 – December 23 | Grumium and Etanin | <ref>{{cite web |url=https://www.interactivestars.com/zodiac_pdf/The_Dragon.pdf |title=The Dragon |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |- | The Serpent | December 24 – December 28 | Alya | <ref>{{cite web |url=https://jumpshare.com/share/ht2Iim8ke1VrgYzEPIFQ |title=The Serpent |publisher=InteractiveStars.com |access-date=2026-09-24 }}</ref> |} == Natal chart conversion methodology == The Lost Zodiac proposes a proportional conversion between a conventional zodiacal position and the corresponding Lost Zodiac period. A placement is first treated as a position within its conventional zodiac sign. That position is converted into a corresponding calendar date and time. The resulting date and time are then located within the relevant Lost Zodiac period. Each listed period includes the full span of every calendar date in its range. The starting date begins at 12:00 a.m., and the ending date concludes at 11:59 p.m. For example, a period ending on February 28 includes the entire day of February 28. In leap years, the Swan period ends at 11:59 p.m. on February 29. For a Lost Zodiac period, the beginning of the period is treated as 0°00′ and the end of the period as 29°59′. The beginning boundary is midnight at the start of the first listed date, while the ending boundary is 11:59 p.m. on the last listed date. The proposed formula is: <math> L = \frac{t-t_0}{t_1-t_0}\times29^\circ 59' </math> where: * <math>L</math> = resulting Lost Zodiac degree * <math>t</math> = the corresponding date and time of the natal placement * <math>t_0</math> = beginning of the Lost Zodiac period * <math>t_1</math> = end boundary of the Lost Zodiac period === Example === A tropical Sun position of Pisces 1°29′ corresponds to February 20, 2009 at approximately 6:25 AM in the example used by the author. The Swan period runs from February 9 through the last day of February, ending on February 28 in a common year or February 29 in a leap year. The position of February 20 at 6:25 AM is approximately 56.34% through the Swan period. Therefore: <math> 29^\circ 59'\times0.5634\approx16^\circ 53' </math> The resulting proposed Lost Zodiac position is therefore: '''Swan 16°53′''' == Reference natal chart == The following conventional natal-chart placements are used as the reference positions for the Lost Zodiac conversion. {| class="wikitable" ! Natal point ! Conventional natal position ! House |- | Sun | Pisces 1°29′ | 1st House |- | Ascendant | Aquarius 29°24′ | — |- | Moon | Capricorn 5°24′ | 11th House |- | Mercury | Aquarius 6°07′ | 12th House |- | Venus | Aries 11°30′ | 2nd House |- | Mars | Aquarius 11°55′ | 12th House |- | Jupiter | Aquarius 10°38′ | 12th House |- | Saturn | Virgo 19°41′ Rx | 7th House |- | Uranus | Pisces 21°25′ | 1st House |- | Neptune | Aquarius 24°13′ | 12th House |- | Pluto | Capricorn 2°47′ | 10th House |- | North Node | Aquarius 8°18′ Rx | 12th House |- | Lilith | Capricorn 5°16′ | 11th House |- | Chiron | Aquarius 21°52′ | 12th House |- | Fortune | Aries 25°29′ | 2nd House |- | Vertex | Virgo 17°15′ | 7th House |- | MC | Sagittarius 8°28′ | — |} == Example: Author's Lost Zodiac natal chart == Using the conversion methodology described above, the author's proposed Lost Zodiac natal placements are: {| class="wikitable" ! Natal point ! Lost Zodiac placement |- | Sun | Swan 16°53′ |- | Ascendant | Swan 14°06′ |- | Moon | Serpent 20°23′ |- | Mercury | Eagle 24°13′ |- | Venus | Pegasus 29°14′ |- | Mars | Dolphin 7°57′ |- | Jupiter | Dolphin 4°27′ |- | Saturn | Cup 1°52′ |- | Uranus | River of Night 28°32′ |- | Neptune | Swan 6°19′ |- | Pluto | Serpent 4°42′ |- | North Node | Eagle 28°35′ |- | Lilith | Serpent 19°35′ |- | Chiron | Swan 2°48′ |- | Fortune | River of Night 18°16′ |- | Vertex | Great Bear 27°04′ |- | MC | Wise Centaur 19°37′ |} gf4vgab61bk0swbe8nogo47jk3gvemj 1 332259 2834851 2026-09-28T12:33:40Z 202 Xup iartkab lakarg 3111500 /* Refinement of webpage content */ new section 2834851 wikitext text/x-wiki == Refinement of webpage content == What do you guys think I should improve next in this webpage I made? [[User:202 Xup iartkab lakarg|202 Xup iartkab lakarg]] ([[User talk:202 Xup iartkab lakarg|discuss]] • [[Special:Contributions/202 Xup iartkab lakarg|contribs]]) 12:33, 28 September 2026 (UTC) o9boicc3v2agi80rtrdtrezxe1vguot 3 332260 2834852 2026-09-28T12:50:19Z MathXplore 2888076 advert1 ([[m:User:ZbVl/VD|Vandoom]]) 2834852 wikitext text/x-wiki == 2026-09-28 == <div class="mw-content-ltr" dir="ltr" style="text-align: left" lang="en">[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Ojiaka|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using <span style="white-space:nowrap">Wikiversity</span> as a "soapbox"]] are not permitted. Take a look at the welcome pages to learn more about <span style="white-space:nowrap">Wikiversity</span>. Thanks. </div><!-- Glow-advert1 @ 1790599816267.6s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:50, 28 September 2026 (UTC) fiyklkv4yl5tr3c93ay89terpoj7tnr 0 332261 2834857 2026-09-28T14:48:04Z Evgenii Arsentev 3110906 New learning resource: three checks before depending on an open-source AI project, with exercises on an open CC BY dataset 2834857 wikitext text/x-wiki {{Short description|How to check an open-source AI project before you build on it, taught with one open dataset of 490 selected projects}} This learning resource is a sub-page of [[Artificial intelligence]]. It is about a question that anyone who picks free AI tools runs into sooner or later: '''a project looks popular today, but will it still be there next month, and are you even allowed to use it?''' It works through one open dataset, the AI Tools Radar, and uses its numbers to practise three quick checks and one habit of mind: a curated list is not a random sample. {{RightTOC}} == Learning objectives == After working through this resource a learner should be able to: * explain what a code repository, a licence and a "last push" are, in plain words; * run three checks on any GitHub project in about a minute; * compute shares from a small table and say what the denominator is; * explain survivorship bias and why a dataset has to keep the projects that disappeared; * say what a curated sample can and cannot tell you. == 1. Words you need == A '''repository''' is the folder where a project's code lives, together with its history. On GitHub each repository has a page that anyone can open. A '''licence''' is the text that says what you may do with the code. Without one, the default in most countries is that you may look at the code but not reuse it. GitHub shows the licence it recognises in the side panel; when it finds a licence file but cannot tell which licence it is, the API reports <code>NOASSERTION</code>. A '''push''' is a new batch of changes uploaded to the repository. The date of the last push is a rough sign of whether anyone is still working on the project. == 2. The dataset == The AI Tools Radar on arsentev.ai selects and reviews open-source AI projects from GitHub, Hugging Face and Hacker News. The first edition of its open dataset covers projects selected between 11 June and 12 September 2026. {| class="wikitable" ! Quantity !! Value |- | Projects selected || 490 (GitHub 422, Hugging Face 63, Hacker News 5) |- | By kind || 427 repositories, 48 Hugging Face models, 15 Hugging Face Spaces |- | Median GitHub stars at the moment of selection || 493 |- | Snapshot date || 12 September 2026 (one snapshot so far) |} [[File:AI tools radar - open-source AI projects selected per month (2026).png|thumb|600px|center|Projects selected per month. September covers only the first 12 days.]] The snapshot was taken on the last day of selection, so for some projects it came the same day they were selected and for others about three months later. Keep this in mind in every exercise below. == 3. Is it still there? == [[File:AI tools radar - selected GitHub repositories available, deleted and archived (2026).png|thumb|600px|center|Status of 427 selected GitHub repositories on 12 September 2026.]] On the snapshot, 416 of the 427 GitHub repositories were available, 10 had been deleted (the API answered 404) and 1 was archived, which means the owner froze it and no longer accepts changes. '''Exercise 1.''' What share of the selected GitHub repositories was no longer usable in the normal way (deleted or archived)? ''(Answer: 11 / 427 ≈ 2.6%.)'' '''Exercise 2.''' Why would it be a mistake to compute this share only from the repositories that the radar still lists on its pages? ''(Hint: a list that quietly drops the projects that vanished can only ever show survivors. The dataset keeps removed repositories with their status for exactly this reason.)'' == 4. Are you allowed to use it? == [[File:AI tools radar - licences of selected GitHub repositories (2026).png|thumb|600px|center|Licences of the 416 available repositories.]] Of the 416 available repositories: MIT 218, Apache-2.0 93, NOASSERTION 43, no licence 36, AGPL-3.0 16, GPL-3.0 8, Unlicense 1, ODbL-1.0 1. '''Exercise 3.''' The dataset reports that 19% of available repositories have no licence or one GitHub does not recognise. Check this from the table. ''(Answer: (43 + 36) / 416 ≈ 19%.)'' '''Exercise 4.''' AGPL-3.0 and GPL-3.0 are licences that allow reuse but attach conditions to it. Find out what the AGPL asks of a company that runs modified code as a web service, and write one sentence on why this matters before you build a product on such a project. == 5. Is anyone working on it? == On the snapshot, 71.6% of the available repositories had a push within the previous 30 days. '''Exercise 5.''' Roughly how many repositories is that? ''(Answer: 0.716 × 416 ≈ 298.)'' '''Exercise 6.''' A project with no push for 30 days is not necessarily dead. Name two reasons why a healthy project might go a month without changes. == 6. The three-check habit == Before you depend on an open-source AI project, open its page and answer three questions: # Does the repository still open, and is it not marked as archived? # Is a known licence named in the side panel, and do its conditions fit what you plan to do? # When was the last push? '''Exercise 7.''' Pick any three AI projects you have seen recommended this month and run the three checks. Record the answers in a table like the ones above. == 7. What a curated sample can and cannot show == The radar's projects are the ones it chose to review, not a random draw from GitHub. So the numbers describe this list, not open-source AI as a whole. '''Exercise 8.''' Give one way the selection could make the licence share look better than on GitHub in general, and one way it could make it look worse. '''Exercise 9.''' The dataset is released as a new version every month with a daily snapshot of each project. Once two editions exist, design a comparison that would show how the share of deleted repositories changes with time since selection. What would you need to hold constant? == Sources == * Arsentev, E. (2026). ''AI Tools Radar: GitHub and Hugging Face projects selected by arsentev.ai'', edition 2026-09-13 [data set]. Zenodo. doi:10.5281/zenodo.22730451. Series (always the latest edition): doi:10.5281/zenodo.22730450. CC BY 4.0. [[Category:Artificial intelligence]] [[Category:Open source]] fndxsvj2mmdipe0gls9ggoo3rxglqfx 2 332262 2834860 2026-09-28T15:50:30Z BajookThug 2920913 Created page with "Lenin wrote ''What Is to Be Done'' in 1902 under conditions of ideological confusion, organizational disintegration, and opportunist retreat within the Russian Social-Democratic movement. The parallels to our present situation are not accidental; they are structural. Wherever the revolutionary vanguard is weak, wherever spontaneity is worshipped and theory is despised, wherever the mass movement is left to drift toward trade-unionist consciousness, the same battles must..." 2834860 wikitext text/x-wiki Lenin wrote ''What Is to Be Done'' in 1902 under conditions of ideological confusion, organizational disintegration, and opportunist retreat within the Russian Social-Democratic movement. The parallels to our present situation are not accidental; they are structural. Wherever the revolutionary vanguard is weak, wherever spontaneity is worshipped and theory is despised, wherever the mass movement is left to drift toward trade-unionist consciousness, the same battles must be fought again. ==''I.'' The Critique of ''"Spontaneity Worship"'': The Root of the Crisis== Lenin begins from a diagnosis of the crisis. The Russian Social-Democratic Party was in a state of "chaos, disintegration, and vacillation".<ref>{{Cite web|url=http://marxism.cass.cn/dsydj/202504/t20250430_5872062.shtml|title=郭超 于海青：列宁解决党内问题的思路与中国共产党的自我革命-马克思主义研究网|website=marxism.cass.cn|language=en|access-date=2026-09-28|quote=19世纪末20世纪初，俄国社会民主党出现了严重的党内问题，党在思想上和组织上陷入了'混乱、瓦解和动摇'。}}</ref> The cause was not merely external repression, though that existed. The cause was internal: the party had been penetrated by a tendency that worshipped the spontaneity of the workers’ movement and denigrated the role of conscious socialist leadership. This tendency, known as "Economism," held that the working class would naturally arrive at revolutionary consciousness through its own economic struggles. The task of the party, on this view, was to tail behind the movement, to support wage demands, to avoid "premature" political struggle. Lenin saw this as a catastrophic error. The fundamental mistake of the Economists, he wrote, was "the worship of spontaneity, the failure to understand that the spontaneity of the masses demands from us Social-Democrats a tremendous amount of consciousness". Why is this an error of principle, not merely of tactics? Because there is no neutral ground in the ideological sphere. "Either the bourgeois ideology, or the socialist ideology. There is no middle course". To worship spontaneity is to surrender the working class to bourgeois ideological domination. The spontaneous workers’ movement, left to itself, tends toward trade-unionism; the struggle for better conditions within the existing system, not the overthrow of that system. Trade-unionism, Lenin insisted, "means precisely the ideological enslavement of the workers by the bourgeoisie". One must therefore grasp the dialectic of spontaneity and consciousness. Spontaneity is the raw material of revolution; it is the awakening of the masses. But raw material is not a finished product. The spontaneous movement must be elevated to conscious revolutionary struggle. This elevation is the task of the vanguard. The more the spontaneous movement grows, the more urgently the party must display "tremendous consciousness" in theory, politics, and organization. ==''II. “Without Revolutionary Theory, There Can Be No Revolutionary Movement”''== Lenin’s most famous formulation from this period is not a slogan but a material analysis of the relationship between theory and practice. The Economists despised theory. They wanted "practical" work, concrete gains, immediate results. Lenin replied that this pseudo-practicality was the most impractical thing imaginable. A movement without theory is a movement without direction. It cannot distinguish friend from enemy, cannot anticipate the turns of struggle, cannot educate its own cadres, cannot maintain its own cohesion. The classical analysis emphasizes three levels of theoretical consciousness required of the party. First, the party must consciously sum up the experience of the socialist movement, distilling lessons from victories and defeats into coherent theory. Second, it must combine this general theory with the concrete conditions of its own country, innovating theoretically rather than mechanically copying the experience of others. Third, it must use this advanced theory to guide social practice, transforming theoretical clarity into organizational and political action. Lenin’s warning is directed not only at those who openly reject theory, but also at those who reduce Marxism to a collection of formulas. The critique of “freedom of criticism” is central here. When opportunists demanded “freedom of criticism,” they were not demanding the freedom to develop revolutionary theory. They were demanding the freedom to abandon it. Lenin saw through this immediately. What they called “freedom of criticism” was in reality “the freedom of the opportunist trend within Social-Democracy, the freedom to transform Social-Democracy into a democratic party of reform, the freedom to introduce bourgeois ideas and bourgeois elements into socialism”. The cadre must therefore be armed with theory and must be ruthless in defending it. Theory is not an ornament. It is the party’s nervous system. Without it, the party cannot see, cannot move, cannot fight. ==''III.'' The Political Struggle: Beyond Trade-Unionism== One of the most important contributions of What Is to Be Done? is its insistence that the economic struggle is not the whole of politics. The Economists argued that the working class should focus on wages, hours, and conditions; the immediate concerns of daily life. Political struggle, they said, was the business of liberals and intellectuals. Lenin rejected this division of labor as a betrayal of the working class. The material fact is that economic struggle alone confines the workers to the horizon of the existing system. It fights for a larger share of the pie without questioning who owns the bakery. Lenin insisted that the party must lead the working class into political struggle: the struggle against autocracy, against the whole apparatus of class rule, against the state itself. “The task,” he wrote, “is to organize comprehensive political exposure”. The party must become the tribune of the people, denouncing every act of oppression, every abuse of power, every instance of exploitation, not only in the factories, but in every sphere of social life. This requires “the most extensive political agitation”. The cadre cannot wait for the workers to spontaneously develop political consciousness. The cadre must bring political consciousness to the workers, must connect every economic grievance to its political root, must transform scattered discontent into organized revolutionary will. This is the work of “consciousness-raising” in the most concrete sense: the patient, persistent, systematic education of the advanced workers, so that they in turn can educate the broader masses. ==''IV.'' The Vanguard Party: Organization as the Weapon of Revolution== Lenin’s most controversial and most misunderstood argument in What Is to Be Done? concerns organization. The Economists preferred loose, decentralized, “democratic” forms. They distrusted centralized leadership. They spoke of “broad democracy” while the party was being destroyed by police repression and internal disorganization. Lenin’s reply was grounded in concrete analysis, not abstract preference. Under conditions of autocratic repression, a loose organization cannot survive. It cannot maintain continuity, cannot preserve its accumulated experience, cannot protect its cadres from infiltration and destruction. The party must therefore be organized on the principle of centralism: a stable core of professional revolutionaries, capable of maintaining the line, preserving the tradition, and adapting to changing circumstances. This does not mean the party is a conspiracy of intellectuals detached from the masses. The professional revolutionary is not a bureaucrat but a cadre: a person whose whole life is dedicated to the revolution, who has mastered theory, who is connected to the workers, who can lead in struggle and educate in quiet moments. The party is the organized expression of the revolutionary will of the class. Without organization, the class is a crowd. With organization, it becomes a force capable of storming heaven. The classical analysis connects this organizational principle to the broader question of leadership. The Economists attacked the idea of a centralized leadership as “anti-democratic,” as a “conspiracy” that substituted itself for the masses. Lenin replied that this was precisely the kind of demagogy that served the enemy. The masses need leadership. The working class needs a vanguard that can clarify its aims, coordinate its struggles, and prevent it from being misled by the bourgeoisie. To refuse this leadership is not democracy; it is abdication. ==''V.'' The Newspaper as Collective Organizer== One of Lenin’s most creative practical proposals in What Is to Be Done? is the role of the all-Russian political newspaper. This is not merely a propaganda instrument. It is a collective organizer. The newspaper serves as a network, connecting scattered local committees into a coherent whole. It provides a common language, a shared analysis, a unified line. It allows the party to intervene in every significant event, to expose every outrage, to draw lessons from every struggle. The classical analysis emphasizes that the newspaper is part of the “ideological leadership” of the party. It is a means of instilling socialist consciousness into the workers’ movement, of countering bourgeois ideological domination, of building the party’s intellectual and moral authority. The cadre must therefore understand that the press is not a luxury but a necessity. It is the instrument through which the vanguard maintains its connection with the masses and its coherence as a fighting force. ==''VI.'' The Lessons for the Worker Today== What are the lessons of What Is to Be Done? for the cadre in our present circumstances? The classical analysis draws several conclusions. First, ideological clarity is the foundation of organizational unity. The party cannot be united if its members do not share a common theoretical understanding and a common political line. Study is not a formality; it is the lifeblood of the vanguard. Second, spontaneity is not enough. The masses will not spontaneously arrive at revolutionary consciousness. The party must lead, educate, and organize. This requires patience, discipline, and a long-term perspective. Third, the economic struggle must be subordinated to the political struggle. The party must fight for reforms, but it must never reduce itself to a reformist organization. Every struggle must be connected to the larger goal of revolutionary transformation. Fourth, organization is not a fetish. Centralism is not an end in itself. It is a weapon adapted to conditions of struggle. The cadre must be flexible in tactics but firm in principle. Fifth, the party must be a school for revolutionaries. It must educate its members not only in theory but in practice. It must cultivate cadres who can lead, who can analyze concrete situations, who can maintain the line under pressure. ==''VII.'' Conclusion: The Vanguard as the Conscious Expression of the Class== Lenin’s What Is to Be Done? is a call to consciousness. It is a rejection of drift, of passivity, of the illusion that history will make itself. It is a demand that the revolutionary vanguard take responsibility for the fate of the movement. The cadre who studies this text must not treat it as a sacred relic but as a living guide. The questions Lenin posed are our questions. The answers he gave must be tested, adapted, and applied to our own conditions. Fix eyes on the task. The spontaneous movement is rising. The crisis of capitalism is deepening. The question is not whether the masses will move, but whether the vanguard will be ready to lead. Organize. Educate. Struggle. The future belongs to the conscious. ==The Three Questions== The questions Lenin posed in ''What Is to Be Done'' are the concrete, urgent questions of revolutionary practice. As Lenin himself stated in the preface, the book was originally intended to analyze three problems raised in his earlier article "From Where to Begin?": the nature and main content of our political agitation, our organizational tasks, and the plan for building an all-Russian fighting organization in various places simultaneously. ===1. What Is the Nature and Main Content of Our Political Agitation?=== This question cuts to the heart of the party's relationship with the working class. The Economists; the Russian opportunists Lenin was combating, reduced political agitation to the economic struggle: wages, hours, factory conditions. They tailed behind the spontaneous workers' movement, limiting their horizons to what could be won within the existing system. Lenin's answer was a fundamental rupture with this trade-unionist consciousness. The materialist analysis begins from a simple observation: the working class, left to itself, develops only trade-union consciousness; an awareness of immediate grievances insufficient for revolutionary action. This is not a moral failing but a structural fact of capitalist society. The spontaneous movement, however militant, remains trapped within the horizon of the existing order. It fights for a larger share of the pie without questioning who owns the bakery. Lenin's answer was therefore that political agitation must be comprehensive and political in the revolutionary sense. It must expose not merely factory owners but the entire apparatus of class rule: the autocracy, the state, the ideological institutions that justify exploitation. The party must become the tribune of the people, denouncing every act of oppression in every sphere of social life. The content of agitation is not the narrow economic grievance but the whole system of oppression. The form is not the factory leaflet alone but the political newspaper, the mass meeting, the organized campaign. The analytical summary emphasizes that this is a question of ideological leadership. The working class does not spontaneously arrive at socialist consciousness; that consciousness must be brought to it from outside the spontaneous movement by the vanguard. To refuse this task, to say that the workers will figure it out themselves, is to abandon them to bourgeois ideological domination. There is no neutral ground: either socialist ideology or bourgeois ideology. The party must choose, and it must fight. ===2. What Are Our Organizational Tasks?=== f the first question concerns consciousness, the second concerns the instrument through which consciousness is organized into material force. Lenin's answer was unambiguous: the party must build a stable, centralized organization of professional revolutionaries. This was not a bureaucratic preference but a material necessity dictated by conditions of repression. The Economists preferred loose, decentralized, "democratic" forms. They spoke of "broad democracy" while the party was being destroyed by police infiltration and internal disorganization. Lenin's analysis of the "handicraft" method; the scattered, amateurish, localized mode of organization, all showed that it was incapable of sustaining a revolutionary struggle against a modern state. A handicraft organization cannot maintain continuity, cannot preserve accumulated experience, cannot protect its cadres, cannot coordinate action across regions. The organizational task, therefore, was to build what Lenin called the "fighting organization of revolutionaries"; a core of cadres whose whole lives were dedicated to the revolution, who had mastered theory, who were connected to the workers, who could lead in struggle and educate in quiet moments. This organization must be centralized to ensure unity of action and secret to survive repression, but it must not be a conspiracy detached from the masses. It must be the organized expression of the revolutionary will of the class. The analytical summary connects this to the question of leadership. The Economists attacked centralized leadership as "anti-democratic," as a "conspiracy" that substituted itself for the masses. Lenin replied that this was demagogy serving the enemy. The masses need leadership. The working class needs a vanguard that can clarify its aims, coordinate its struggles, and prevent it from being misled by the bourgeoisie. To refuse this leadership is not democracy; it is abdication. The party must be organized to lead, not to follow. ===3. What Is the Plan for Building an All-Russian Fighting Organization?=== The third question is the practical synthesis of the first two. It asks: how do we actually build this organization? Lenin's answer was the all-Russian political newspaper, conceived not merely as a propaganda instrument but as a collective organizer. The newspaper serves as a network, connecting scattered local committees into a coherent whole. It provides a common language, a shared analysis, a unified line. It allows the party to intervene in every significant event, to expose every outrage, to draw lessons from every struggle. Through the newspaper, the party maintains ideological leadership over the movement and builds the organizational infrastructure of the vanguard. But the newspaper was only one element of the plan. Lenin's broader conception was the simultaneous construction of local organizations and a central leadership, bound together by the newspaper and by a common political line. The plan required cadres in every region, connected to the workers, capable of independent initiative but loyal to the center. It required a division of labor between agitators, organizers, and writers, all integrated into a single fighting force. The analytical summary emphasizes that this is not a technical blueprint but a strategic conception of party building. The question "what is to be done?" is answered concretely: build the newspaper, build the local committees, build the central leadership, and connect them all through a shared theoretical and political framework. The plan is not utopian; it is the practical application of the first two answers. Without political agitation of the right kind, there is nothing to organize. Without organization, the agitation disperses into the wind. The newspaper and the fighting organization are the means by which consciousness becomes power. ===The Unity of the Three Questions=== These three questions are not separate. They are three moments in a single dialectical process. The first establishes the content of revolutionary work: comprehensive political agitation aimed at socialist consciousness. The second establishes the form of revolutionary work: the centralized organization of professional revolutionaries. The third establishes the method of connecting content and form: the all-Russian newspaper and the simultaneous construction of local and central organizations. The questions Lenin posed are our questions because the fundamental problem remains the same: how does the working class, under conditions of bourgeois ideological domination, develop revolutionary consciousness and build the organizational capacity to seize power? The answers Lenin gave must be tested against our conditions, but the questions themselves are inescapable. To avoid them is to abandon the revolutionary project. To answer them is to build the vanguard. The choice is not academic. It is the choice between revolution and reformism, between the conscious leadership of the working class and the tailing behind spontaneity, between the fighting organization and the scattered group. Lenin's questions are our questions because they are the questions of every revolutionary movement. The answers must be worked out in practice, but the questions must first be grasped in theory. This is the task of the cadre: to study, to organize, to fight, and to answer Lenin's questions with the only answer that matters; RA. b4i9e38v3nrfvo3xwoh9xl3yzvje3l3 2834861 2834860 2026-09-28T15:51:34Z BajookThug 2920913 2834861 wikitext text/x-wiki ''(summary and analysis of Lenin's book)'' Lenin wrote ''What Is to Be Done'' in 1902 under conditions of ideological confusion, organizational disintegration, and opportunist retreat within the Russian Social-Democratic movement. The parallels to our present situation are not accidental; they are structural. Wherever the revolutionary vanguard is weak, wherever spontaneity is worshipped and theory is despised, wherever the mass movement is left to drift toward trade-unionist consciousness, the same battles must be fought again. ==''I.'' The Critique of ''"Spontaneity Worship"'': The Root of the Crisis== Lenin begins from a diagnosis of the crisis. The Russian Social-Democratic Party was in a state of "chaos, disintegration, and vacillation".<ref>{{Cite web|url=http://marxism.cass.cn/dsydj/202504/t20250430_5872062.shtml|title=郭超 于海青：列宁解决党内问题的思路与中国共产党的自我革命-马克思主义研究网|website=marxism.cass.cn|language=en|access-date=2026-09-28|quote=19世纪末20世纪初，俄国社会民主党出现了严重的党内问题，党在思想上和组织上陷入了'混乱、瓦解和动摇'。}}</ref> The cause was not merely external repression, though that existed. The cause was internal: the party had been penetrated by a tendency that worshipped the spontaneity of the workers’ movement and denigrated the role of conscious socialist leadership. This tendency, known as "Economism," held that the working class would naturally arrive at revolutionary consciousness through its own economic struggles. The task of the party, on this view, was to tail behind the movement, to support wage demands, to avoid "premature" political struggle. Lenin saw this as a catastrophic error. The fundamental mistake of the Economists, he wrote, was "the worship of spontaneity, the failure to understand that the spontaneity of the masses demands from us Social-Democrats a tremendous amount of consciousness". Why is this an error of principle, not merely of tactics? Because there is no neutral ground in the ideological sphere. "Either the bourgeois ideology, or the socialist ideology. There is no middle course". To worship spontaneity is to surrender the working class to bourgeois ideological domination. The spontaneous workers’ movement, left to itself, tends toward trade-unionism; the struggle for better conditions within the existing system, not the overthrow of that system. Trade-unionism, Lenin insisted, "means precisely the ideological enslavement of the workers by the bourgeoisie". One must therefore grasp the dialectic of spontaneity and consciousness. Spontaneity is the raw material of revolution; it is the awakening of the masses. But raw material is not a finished product. The spontaneous movement must be elevated to conscious revolutionary struggle. This elevation is the task of the vanguard. The more the spontaneous movement grows, the more urgently the party must display "tremendous consciousness" in theory, politics, and organization. ==''II. “Without Revolutionary Theory, There Can Be No Revolutionary Movement”''== Lenin’s most famous formulation from this period is not a slogan but a material analysis of the relationship between theory and practice. The Economists despised theory. They wanted "practical" work, concrete gains, immediate results. Lenin replied that this pseudo-practicality was the most impractical thing imaginable. A movement without theory is a movement without direction. It cannot distinguish friend from enemy, cannot anticipate the turns of struggle, cannot educate its own cadres, cannot maintain its own cohesion. The classical analysis emphasizes three levels of theoretical consciousness required of the party. First, the party must consciously sum up the experience of the socialist movement, distilling lessons from victories and defeats into coherent theory. Second, it must combine this general theory with the concrete conditions of its own country, innovating theoretically rather than mechanically copying the experience of others. Third, it must use this advanced theory to guide social practice, transforming theoretical clarity into organizational and political action. Lenin’s warning is directed not only at those who openly reject theory, but also at those who reduce Marxism to a collection of formulas. The critique of “freedom of criticism” is central here. When opportunists demanded “freedom of criticism,” they were not demanding the freedom to develop revolutionary theory. They were demanding the freedom to abandon it. Lenin saw through this immediately. What they called “freedom of criticism” was in reality “the freedom of the opportunist trend within Social-Democracy, the freedom to transform Social-Democracy into a democratic party of reform, the freedom to introduce bourgeois ideas and bourgeois elements into socialism”. The cadre must therefore be armed with theory and must be ruthless in defending it. Theory is not an ornament. It is the party’s nervous system. Without it, the party cannot see, cannot move, cannot fight. ==''III.'' The Political Struggle: Beyond Trade-Unionism== One of the most important contributions of What Is to Be Done? is its insistence that the economic struggle is not the whole of politics. The Economists argued that the working class should focus on wages, hours, and conditions; the immediate concerns of daily life. Political struggle, they said, was the business of liberals and intellectuals. Lenin rejected this division of labor as a betrayal of the working class. The material fact is that economic struggle alone confines the workers to the horizon of the existing system. It fights for a larger share of the pie without questioning who owns the bakery. Lenin insisted that the party must lead the working class into political struggle: the struggle against autocracy, against the whole apparatus of class rule, against the state itself. “The task,” he wrote, “is to organize comprehensive political exposure”. The party must become the tribune of the people, denouncing every act of oppression, every abuse of power, every instance of exploitation, not only in the factories, but in every sphere of social life. This requires “the most extensive political agitation”. The cadre cannot wait for the workers to spontaneously develop political consciousness. The cadre must bring political consciousness to the workers, must connect every economic grievance to its political root, must transform scattered discontent into organized revolutionary will. This is the work of “consciousness-raising” in the most concrete sense: the patient, persistent, systematic education of the advanced workers, so that they in turn can educate the broader masses. ==''IV.'' The Vanguard Party: Organization as the Weapon of Revolution== Lenin’s most controversial and most misunderstood argument in What Is to Be Done? concerns organization. The Economists preferred loose, decentralized, “democratic” forms. They distrusted centralized leadership. They spoke of “broad democracy” while the party was being destroyed by police repression and internal disorganization. Lenin’s reply was grounded in concrete analysis, not abstract preference. Under conditions of autocratic repression, a loose organization cannot survive. It cannot maintain continuity, cannot preserve its accumulated experience, cannot protect its cadres from infiltration and destruction. The party must therefore be organized on the principle of centralism: a stable core of professional revolutionaries, capable of maintaining the line, preserving the tradition, and adapting to changing circumstances. This does not mean the party is a conspiracy of intellectuals detached from the masses. The professional revolutionary is not a bureaucrat but a cadre: a person whose whole life is dedicated to the revolution, who has mastered theory, who is connected to the workers, who can lead in struggle and educate in quiet moments. The party is the organized expression of the revolutionary will of the class. Without organization, the class is a crowd. With organization, it becomes a force capable of storming heaven. The classical analysis connects this organizational principle to the broader question of leadership. The Economists attacked the idea of a centralized leadership as “anti-democratic,” as a “conspiracy” that substituted itself for the masses. Lenin replied that this was precisely the kind of demagogy that served the enemy. The masses need leadership. The working class needs a vanguard that can clarify its aims, coordinate its struggles, and prevent it from being misled by the bourgeoisie. To refuse this leadership is not democracy; it is abdication. ==''V.'' The Newspaper as Collective Organizer== One of Lenin’s most creative practical proposals in What Is to Be Done? is the role of the all-Russian political newspaper. This is not merely a propaganda instrument. It is a collective organizer. The newspaper serves as a network, connecting scattered local committees into a coherent whole. It provides a common language, a shared analysis, a unified line. It allows the party to intervene in every significant event, to expose every outrage, to draw lessons from every struggle. The classical analysis emphasizes that the newspaper is part of the “ideological leadership” of the party. It is a means of instilling socialist consciousness into the workers’ movement, of countering bourgeois ideological domination, of building the party’s intellectual and moral authority. The cadre must therefore understand that the press is not a luxury but a necessity. It is the instrument through which the vanguard maintains its connection with the masses and its coherence as a fighting force. ==''VI.'' The Lessons for the Worker Today== What are the lessons of What Is to Be Done? for the cadre in our present circumstances? The classical analysis draws several conclusions. First, ideological clarity is the foundation of organizational unity. The party cannot be united if its members do not share a common theoretical understanding and a common political line. Study is not a formality; it is the lifeblood of the vanguard. Second, spontaneity is not enough. The masses will not spontaneously arrive at revolutionary consciousness. The party must lead, educate, and organize. This requires patience, discipline, and a long-term perspective. Third, the economic struggle must be subordinated to the political struggle. The party must fight for reforms, but it must never reduce itself to a reformist organization. Every struggle must be connected to the larger goal of revolutionary transformation. Fourth, organization is not a fetish. Centralism is not an end in itself. It is a weapon adapted to conditions of struggle. The cadre must be flexible in tactics but firm in principle. Fifth, the party must be a school for revolutionaries. It must educate its members not only in theory but in practice. It must cultivate cadres who can lead, who can analyze concrete situations, who can maintain the line under pressure. ==''VII.'' Conclusion: The Vanguard as the Conscious Expression of the Class== Lenin’s What Is to Be Done? is a call to consciousness. It is a rejection of drift, of passivity, of the illusion that history will make itself. It is a demand that the revolutionary vanguard take responsibility for the fate of the movement. The cadre who studies this text must not treat it as a sacred relic but as a living guide. The questions Lenin posed are our questions. The answers he gave must be tested, adapted, and applied to our own conditions. Fix eyes on the task. The spontaneous movement is rising. The crisis of capitalism is deepening. The question is not whether the masses will move, but whether the vanguard will be ready to lead. Organize. Educate. Struggle. The future belongs to the conscious. ==The Three Questions== The questions Lenin posed in ''What Is to Be Done'' are the concrete, urgent questions of revolutionary practice. As Lenin himself stated in the preface, the book was originally intended to analyze three problems raised in his earlier article "From Where to Begin?": the nature and main content of our political agitation, our organizational tasks, and the plan for building an all-Russian fighting organization in various places simultaneously. ===1. What Is the Nature and Main Content of Our Political Agitation?=== This question cuts to the heart of the party's relationship with the working class. The Economists; the Russian opportunists Lenin was combating, reduced political agitation to the economic struggle: wages, hours, factory conditions. They tailed behind the spontaneous workers' movement, limiting their horizons to what could be won within the existing system. Lenin's answer was a fundamental rupture with this trade-unionist consciousness. The materialist analysis begins from a simple observation: the working class, left to itself, develops only trade-union consciousness; an awareness of immediate grievances insufficient for revolutionary action. This is not a moral failing but a structural fact of capitalist society. The spontaneous movement, however militant, remains trapped within the horizon of the existing order. It fights for a larger share of the pie without questioning who owns the bakery. Lenin's answer was therefore that political agitation must be comprehensive and political in the revolutionary sense. It must expose not merely factory owners but the entire apparatus of class rule: the autocracy, the state, the ideological institutions that justify exploitation. The party must become the tribune of the people, denouncing every act of oppression in every sphere of social life. The content of agitation is not the narrow economic grievance but the whole system of oppression. The form is not the factory leaflet alone but the political newspaper, the mass meeting, the organized campaign. The analytical summary emphasizes that this is a question of ideological leadership. The working class does not spontaneously arrive at socialist consciousness; that consciousness must be brought to it from outside the spontaneous movement by the vanguard. To refuse this task, to say that the workers will figure it out themselves, is to abandon them to bourgeois ideological domination. There is no neutral ground: either socialist ideology or bourgeois ideology. The party must choose, and it must fight. ===2. What Are Our Organizational Tasks?=== f the first question concerns consciousness, the second concerns the instrument through which consciousness is organized into material force. Lenin's answer was unambiguous: the party must build a stable, centralized organization of professional revolutionaries. This was not a bureaucratic preference but a material necessity dictated by conditions of repression. The Economists preferred loose, decentralized, "democratic" forms. They spoke of "broad democracy" while the party was being destroyed by police infiltration and internal disorganization. Lenin's analysis of the "handicraft" method; the scattered, amateurish, localized mode of organization, all showed that it was incapable of sustaining a revolutionary struggle against a modern state. A handicraft organization cannot maintain continuity, cannot preserve accumulated experience, cannot protect its cadres, cannot coordinate action across regions. The organizational task, therefore, was to build what Lenin called the "fighting organization of revolutionaries"; a core of cadres whose whole lives were dedicated to the revolution, who had mastered theory, who were connected to the workers, who could lead in struggle and educate in quiet moments. This organization must be centralized to ensure unity of action and secret to survive repression, but it must not be a conspiracy detached from the masses. It must be the organized expression of the revolutionary will of the class. The analytical summary connects this to the question of leadership. The Economists attacked centralized leadership as "anti-democratic," as a "conspiracy" that substituted itself for the masses. Lenin replied that this was demagogy serving the enemy. The masses need leadership. The working class needs a vanguard that can clarify its aims, coordinate its struggles, and prevent it from being misled by the bourgeoisie. To refuse this leadership is not democracy; it is abdication. The party must be organized to lead, not to follow. ===3. What Is the Plan for Building an All-Russian Fighting Organization?=== The third question is the practical synthesis of the first two. It asks: how do we actually build this organization? Lenin's answer was the all-Russian political newspaper, conceived not merely as a propaganda instrument but as a collective organizer. The newspaper serves as a network, connecting scattered local committees into a coherent whole. It provides a common language, a shared analysis, a unified line. It allows the party to intervene in every significant event, to expose every outrage, to draw lessons from every struggle. Through the newspaper, the party maintains ideological leadership over the movement and builds the organizational infrastructure of the vanguard. But the newspaper was only one element of the plan. Lenin's broader conception was the simultaneous construction of local organizations and a central leadership, bound together by the newspaper and by a common political line. The plan required cadres in every region, connected to the workers, capable of independent initiative but loyal to the center. It required a division of labor between agitators, organizers, and writers, all integrated into a single fighting force. The analytical summary emphasizes that this is not a technical blueprint but a strategic conception of party building. The question "what is to be done?" is answered concretely: build the newspaper, build the local committees, build the central leadership, and connect them all through a shared theoretical and political framework. The plan is not utopian; it is the practical application of the first two answers. Without political agitation of the right kind, there is nothing to organize. Without organization, the agitation disperses into the wind. The newspaper and the fighting organization are the means by which consciousness becomes power. ===The Unity of the Three Questions=== These three questions are not separate. They are three moments in a single dialectical process. The first establishes the content of revolutionary work: comprehensive political agitation aimed at socialist consciousness. The second establishes the form of revolutionary work: the centralized organization of professional revolutionaries. The third establishes the method of connecting content and form: the all-Russian newspaper and the simultaneous construction of local and central organizations. The questions Lenin posed are our questions because the fundamental problem remains the same: how does the working class, under conditions of bourgeois ideological domination, develop revolutionary consciousness and build the organizational capacity to seize power? The answers Lenin gave must be tested against our conditions, but the questions themselves are inescapable. To avoid them is to abandon the revolutionary project. To answer them is to build the vanguard. The choice is not academic. It is the choice between revolution and reformism, between the conscious leadership of the working class and the tailing behind spontaneity, between the fighting organization and the scattered group. Lenin's questions are our questions because they are the questions of every revolutionary movement. The answers must be worked out in practice, but the questions must first be grasped in theory. This is the task of the cadre: to study, to organize, to fight, and to answer Lenin's questions with the only answer that matters; RA. 7q2s4s86dyfzzh13mndr45lqxturunk 2 332264 2834869 2026-09-28T17:39:19Z EOA bdargham 3111769 Created page with "{{user page}} Hello. I am Bahaa Budargham, an independent researcher based in Montreal, Canada. I work on the Engineering of Alphabets (EOA), a research program studying deterministic symbolic sequences generated from phonetic encodings of letter names. My current work is on factor complexity, substitution structure, and limiting ratios of these sequences. I am seeking collaboration with researchers in symbolic dynamics, combinatorics on words, and formal languages...." 2834869 wikitext text/x-wiki {{user page}} Hello. I am Bahaa Budargham, an independent researcher based in Montreal, Canada. I work on the Engineering of Alphabets (EOA), a research program studying deterministic symbolic sequences generated from phonetic encodings of letter names. My current work is on factor complexity, substitution structure, and limiting ratios of these sequences. I am seeking collaboration with researchers in symbolic dynamics, combinatorics on words, and formal languages. Contact: bdarghamneurolabs@gmail.com 2xlnod8p44plb60kp05cnvvy417stkc 2834870 2834869 2026-09-28T17:40:01Z EOA bdargham 3111769 2834870 wikitext text/x-wiki {{user page}} Hello. I am Bahaa Budargham, an independent researcher based in Montreal, Canada. I work on the Engineering of Alphabets (EOA), a research program studying deterministic symbolic sequences generated from phonetic encodings of letter names. My current work is on factor complexity, substitution structure, and limiting ratios of these sequences. I am seeking collaboration with researchers in symbolic dynamics, combinatorics on words, and formal languages. i6p5ig9j8lxxxj22gmq658831mcig4m 0 332266 2834878 2026-09-28T19:27:35Z The Citer 3110681 Created page with "The '''lycophytes''', are a phylum of tracheophytes that reproduce with spores.<ref>{{Cite web|url=https://www.britannica.com/plant/lycophyte|title=Lycophyte &#124; Definition, Taxonomy, Characteristics, Examples, & Facts |website=Britannica.com|access-date=May 24, 2026}}</ref> When broadly circumscribed, they include the clubmosses. They are sometimes placed in a division '''Lycopodiophyta''' or '''Lycophyta''' or in a subdivision '''Lycopodiophytina'''. The scientific..." 2834878 wikitext text/x-wiki The '''lycophytes''', are a phylum of tracheophytes that reproduce with spores.<ref>{{Cite web|url=https://www.britannica.com/plant/lycophyte|title=Lycophyte &#124; Definition, Taxonomy, Characteristics, Examples, & Facts |website=Britannica.com|access-date=May 24, 2026}}</ref> When broadly circumscribed, they include the clubmosses. They are sometimes placed in a division '''Lycopodiophyta''' or '''Lycophyta''' or in a subdivision '''Lycopodiophytina'''. The scientific names and the informal English names used for this group of plants are ambiguous. For example, "Lycopodiophyta" and the shorter "Lycophyta" as well as the informal "lycophyte" may be used to include the extinct zosterophylls or to exclude them. ==Information== Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living ==Evolutionary History== '''Lycophytes''' are one of the oldest lineages of extant (living) vascular plants; the group contains extinct plants that have been dated from the [[WikiSilurian]] (ca. 425 million years ago).<ref>{{cite journal|last1=Rickards|first1=R.B.|title=The age of the earliest club mosses: the Silurian Baragwanathia flora in Victoria, Australia|journal=Geological Magazine|date=2000|volume=137|issue=2|pages=207–209|doi=10.1017/s0016756800003800|bibcode=2000GeoM..137..207R|s2cid=131287538}}</ref><ref name="isbn0-19-850065-3">{{cite book |author1=McElwain, Jenny C. |author2=Willis, K. G. |author3=Willis, Kathy |author4=McElwain, J. C. | title = The evolution of plants | publisher = Oxford University Press | location = Oxford [Oxfordshire] | year = 2002 | isbn = 978-0-19-850065-0 }}</ref> Lycophytes were some of the dominating plant species of the [[Carboniferous]] period, and included the tree-like [[Lepidodendrales]], some of which grew over {{Convert|40|m|ft}} in height, although extant lycophytes are relatively small plants.<ref>{{Cite book|title=Biology and evolution of ferns and lycophytes|last1=Ranker|first1=T. A.|last2=Hauler|first2=C. H.|publisher=Cambridge University Press|year=2008|location=Cambridge}}</ref> oyxjyb91cl510jk3acmvhgqox3tbcty 2834885 2834878 2026-09-28T19:47:09Z The Citer 3110681 This will make it educational! 2834885 wikitext text/x-wiki [[File:Lycophyta.png|thumb|300x300px|These are Lycophytes.]] The '''lycophytes''', are a phylum of tracheophytes that reproduce with spores.<ref>{{Cite web|url=https://www.britannica.com/plant/lycophyte|title=Lycophyte &#124; Definition, Taxonomy, Characteristics, Examples, & Facts |website=Britannica.com|access-date=May 24, 2026}}</ref> When broadly circumscribed, they include the clubmosses. They are sometimes placed in a division '''Lycopodiophyta''' or '''Lycophyta''' or in a subdivision '''Lycopodiophytina'''. The scientific names and the informal English names used for this group of plants are ambiguous. For example, "Lycopodiophyta" and the shorter "Lycophyta" as well as the informal "lycophyte" may be used to include the extinct zosterophylls or to exclude them. ==Information== Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living ==Evolutionary History== '''Lycophytes''' are one of the oldest lineages of extant (living) vascular plants; the group contains extinct plants that have been dated from the [[Wikipedia:Silurian|Silurian]] (ca. 425 million years ago).<ref>{{cite journal|last1=Rickards|first1=R.B.|title=The age of the earliest club mosses: the Silurian Baragwanathia flora in Victoria, Australia|journal=Geological Magazine|date=2000|volume=137|issue=2|pages=207–209|doi=10.1017/s0016756800003800|bibcode=2000GeoM..137..207R|s2cid=131287538}}</ref><ref name="isbn0-19-850065-3">{{cite book |author1=McElwain, Jenny C. |author2=Willis, K. G. |author3=Willis, Kathy |author4=McElwain, J. C. | title = The evolution of plants | publisher = Oxford University Press | location = Oxford [Oxfordshire] | year = 2002 | isbn = 978-0-19-850065-0 }}</ref> Lycophytes were some of the dominating plant species of the [[Wikipedia:Carboniferous|Carboniferous]] period, and included the tree-like Lepidodendrales, some of which grew over {{Convert|40|m|ft}} in height, although extant lycophytes are relatively small plants.<ref>{{Cite book|title=Biology and evolution of ferns and lycophytes|last1=Ranker|first1=T. A.|last2=Hauler|first2=C. H.|publisher=Cambridge University Press|year=2008|location=Cambridge}}</ref> ==Classes== Lycopodiopsida – †Zosterophyllopsida ===Genera & Orders ''Incertae sedis''=== †Drepanophycales – †Nothia ==Phylogeny== {{clade |label1=Lycopodiopsida |1={{clade |label1= |1={{clade |label1= |1=Lycopodiales [[File:Lycopodium annotinum1.jpg|60px]] |2=†Drepanophycales [[File:Asteroxylon kz02.jpg|60px]] }} |2={{clade |label1= |1=Selaginellales [[File:Selaginella flabellata.jpg|60px]] |2={{clade |label1= |1=†Lepidodendrales [[File:Lepidophloios reconstrucción.jpg|60px]] |2={{clade |label1= |1=†Pleuromeiales [[File:Pleuromeia restoration.png|60x60px]] |2=Isoetales [[File:Isoetes echinospora 74924873 (4x3).jpg|60px]] }} }} }} }} }} ==References== [[Category:Plants]] [[Category:Taxonomy]] [[Category:Botany]] [[Category:Biology]] 3tphebv5o1oa7r9hry83f35c8wrww7h 2834888 2834885 2026-09-28T19:52:59Z The Citer 3110681 /* Genera & Orders Incertae sedis */ 2834888 wikitext text/x-wiki [[File:Lycophyta.png|thumb|300x300px|These are Lycophytes.]] The '''lycophytes''', are a phylum of tracheophytes that reproduce with spores.<ref>{{Cite web|url=https://www.britannica.com/plant/lycophyte|title=Lycophyte &#124; Definition, Taxonomy, Characteristics, Examples, & Facts |website=Britannica.com|access-date=May 24, 2026}}</ref> When broadly circumscribed, they include the clubmosses. They are sometimes placed in a division '''Lycopodiophyta''' or '''Lycophyta''' or in a subdivision '''Lycopodiophytina'''. The scientific names and the informal English names used for this group of plants are ambiguous. For example, "Lycopodiophyta" and the shorter "Lycophyta" as well as the informal "lycophyte" may be used to include the extinct zosterophylls or to exclude them. ==Information== Name Meaning: Lycopodium-like plants, wolf plant English Common Name: Clubmosses, spikemosses Major distinguishing characteristics: Microphyll leaves, vascular system Approximate number of species described: 1290 living ==Evolutionary History== '''Lycophytes''' are one of the oldest lineages of extant (living) vascular plants; the group contains extinct plants that have been dated from the [[Wikipedia:Silurian|Silurian]] (ca. 425 million years ago).<ref>{{cite journal|last1=Rickards|first1=R.B.|title=The age of the earliest club mosses: the Silurian Baragwanathia flora in Victoria, Australia|journal=Geological Magazine|date=2000|volume=137|issue=2|pages=207–209|doi=10.1017/s0016756800003800|bibcode=2000GeoM..137..207R|s2cid=131287538}}</ref><ref name="isbn0-19-850065-3">{{cite book |author1=McElwain, Jenny C. |author2=Willis, K. G. |author3=Willis, Kathy |author4=McElwain, J. C. | title = The evolution of plants | publisher = Oxford University Press | location = Oxford [Oxfordshire] | year = 2002 | isbn = 978-0-19-850065-0 }}</ref> Lycophytes were some of the dominating plant species of the [[Wikipedia:Carboniferous|Carboniferous]] period, and included the tree-like Lepidodendrales, some of which grew over {{Convert|40|m|ft}} in height, although extant lycophytes are relatively small plants.<ref>{{Cite book|title=Biology and evolution of ferns and lycophytes|last1=Ranker|first1=T. A.|last2=Hauler|first2=C. H.|publisher=Cambridge University Press|year=2008|location=Cambridge}}</ref> ==Classes== Lycopodiopsida – †Zosterophyllopsida ===Genera & Orders ''Incertae sedis''=== †Drepanophycales – †''Nothia'' ==Phylogeny== {{clade |label1=Lycopodiopsida |1={{clade |label1= |1={{clade |label1= |1=Lycopodiales [[File:Lycopodium annotinum1.jpg|60px]] |2=†Drepanophycales [[File:Asteroxylon kz02.jpg|60px]] }} |2={{clade |label1= |1=Selaginellales [[File:Selaginella flabellata.jpg|60px]] |2={{clade |label1= |1=†Lepidodendrales [[File:Lepidophloios reconstrucción.jpg|60px]] |2={{clade |label1= |1=†Pleuromeiales [[File:Pleuromeia restoration.png|60x60px]] |2=Isoetales [[File:Isoetes echinospora 74924873 (4x3).jpg|60px]] }} }} }} }} }} ==References== [[Category:Plants]] [[Category:Taxonomy]] [[Category:Botany]] [[Category:Biology]] i8oefkj164rf84606b08enzb0czi16j 3 332268 2834939 2026-09-29T00:43:37Z Jtneill 10242 Welcome 2834939 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Evgenii Arsentev!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. 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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:43, 29 September 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} odlhts312bcque6osdrc9p7rplhas6d 1 332269 2834962 2026-09-29T03:57:08Z Jtneill 10242 Topic development feedback 2834962 wikitext text/x-wiki == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3252175|Hi [[User:U3252175|U3252175]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:57, 29 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded, missing, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 3-level --> # Overly complicated/messy [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Headings should be more specific (e.g., "Background" is too general; background about what?) <!-- Other ---> # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings <!-- Remove --> # Remove genAI heading - see using genAI guidelines for the unit and as discussed in tutorials <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Insufficient # Needs scenario, image, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. |4= <!-- Key points--> <!-- Overall --> # Insufficient development mainly due to lack of sufficient citation of peer-reviewed theory and research # Provide more detailed edit summaries <!-- Theory and research --> # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- GenAI ---> # If these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]], this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Placeholder use of scenarios/examples/case studies <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Insufficient; 4 out of 6 minimum references used for topic development <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Citations --> # All references need in-text citation <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## use dois where available instead of other links ## remove "Retrieved from ..." ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. |8= <!-- Resources --> <!-- See also --> # See also ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} f9arrns41d2x0ji5e3d5ivlo5ucjukv 2834966 2834962 2026-09-29T04:48:29Z Jtneill 10242 2834966 wikitext text/x-wiki == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3252175|Hi [[User:U3252175|U3252175]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:57, 29 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded, missing, and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 3-level --> # Overly complicated/messy [[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Headings should be more specific (e.g., "Background" is too general; background about what?) <!-- Other ---> # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings <!-- Remove --> # Remove genAI heading - see using genAI guidelines for the unit and as discussed in tutorials <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? 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The goal is to identify and use the best academic theory and research about this topic. |8= <!-- Resources --> <!-- See also --> # See also ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant Wikiversity book chapters and Wikipedia resources about this topic <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} jq1dds3h9q0x4r0ysjbyt8crut2qqg9 3 332270 2834964 2026-09-29T04:38:43Z Jtneill 10242 Welcome 2834964 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], U3252175!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. 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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:38, 29 September 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 22s2f3oy5h5i49wn79358bm2opxui2c 0 332271 2835018 2026-09-29T10:21:47Z Dick Bos 24466 first lines: Jevons 2835018 wikitext text/x-wiki {{0% done}} An important "forerunner" of neo-classical economics was [[:w:en:William Stanley Jevons|William Stanley Jevons]] (1835–1882).<ref name=Stiglitz>G.J. Stiglitz 1946 - Production and Distribution Theories. New York, pp. 13, 135.<!-- check this reference --></ref> The Marginal Revolution is therefore sometimes called "the Jevonian Revolution."{{sfn|Dobb|1973|p=166 f}} Jevons outlined the concept of marginal utility in 1862. He was very much aware of the necessity to reshunt the "car of economic science" which [[David Ricardo]] had directed "onto a wrong line".<ref name=line>Preface to 2nd ed. of ''Theory of Political Economy'' (1879)<!-- check this reference --> as cited by {{harvnb|Dobb|1973|p=166}}</ref> == References == {{references}} == Sources == *{{cite book | last = Dobb | first = Maurice | date = 1973 | title = Theories of Value and Distribution Since Adam Smith: Ideology and Economic Theory | location = London | publisher = Cambridge University Press | ISBN = 0 521 20100 4 | url = https://archive.org/details/theoriesofvalued0000dobb | url-access = registration | ref = harv }} 8ti9hygvfqxkz0jmquun6g6p0zg9gtm 0 332272 2835019 2026-09-29T10:21:54Z MrBerard 3111808 Initial content migrated from https://en.wikibooks.org/wiki/A-level_Computing_AQA_Paper_1_Skeleton_program_2027 2835019 wikitext text/x-wiki This year Skeleton Program is called "Pile O' Tiles", a two player (arguably multiplayer, see below) version of the classic match-2 memory game.</blockquote> == From AQA's preliminary material document: == <blockquote>Piles O’ Tiles is a tile matching memory game. The board consists of a two-dimensional grid containing piles of tiles. The tiles are face-down so players cannot see what is on the tiles. Each tile has a symbol (an uppercase letter) and a points value. Players take it in turn to choose two piles. The top tile of those two piles is turned over so that all players can see what is on those tiles. If the two tiles have the same symbol on them then they are removed from the piles and the points for the two tiles are added to the player’s score. The two tiles are then added to a discard. If the two tiles do not have the same symbol on them then they are turned face-down again and put back on the piles they came from. When the game ends the player with the highest score is the winner. In the Skeleton Program the board is shown as a two-dimensional grid with x and y coordinates. Each pile is shown as an integer value that denotes the number of tiles currently in the pile.</blockquote> Then a diagram showing the game's display, with a 6x6 board, rendered as a grid with the origin in the bottom left, row numbers on the left, and column numbers on the bottom<blockquote>In the Skeleton Program there is a default version of the game that uses a 6×6 grid. It is also possible to load a game from a file. The Data File game1.txt contains a game that can be loaded instead of using the default game. </blockquote>The AQA doc has the content of game1.txt, but we can do better. == Annotated game1.txt == 16 # Grid size in Nsquares - the grid is the a square of side sqrt of that 2,A,1,E,1,H,3,J,2 # Those are piles (of tiles) 0,E,1,Q,4,K,4,J,2,J,2 # arranged linearly (here, 16 of them) 0,O,1,E,1,H,3 # the first element is the bonus for the tile, 0,O,1,E,1,H,3 # then pairs of letters, score 0,A,1,E,1,H,3 0,A,1,H,1,H,3 0,E,1,E,1,H,3 0,A,1,Q,4,F,3 0,U,1,E,1,H,3 0,A,1,E,1,H,3 4,U,5,E,1,K,4 2,A,1,K,4,H,3 0,A,1,E,1,K,1 0,R,1,K,3,H,3 0,R,1,F,3,K,7 0,A,1,E,1,H,3 2 # Number of players Boris,0 # player name, score Poppy,0 # Discard, in pairs of letters, score - here it's empty 15 # max score 0 # whose turn it is (0 is player 1, Boris) 0 # turns since the the last winning move (matching tiles) by any player # Gamefiles are saves of the gamestate, but here this is the start of a game. == Section D mark distribution comparison == Mark distribution for this year: '''- TK''' (please fill in if you have the EAD) These marks include the screen capture(s), so the marks for the coding will be 1-2 marks lower. * The 2026 paper 1 contained 4 questions: a 5 mark, an 8 mark and two 12 mark questions. * The 2025 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2024 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2023 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 10 mark and one 13 mark question. * The 2022 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 11 mark and one 13 mark question. * The 2021 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 9 mark and one 14 mark question. * The 2020 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 11 mark and one 12 mark question. * The 2019 paper 1 contained 4 questions: a 5 mark, an 8 mark, a 9 mark and one 13 mark question. * The 2018 paper 1 contained 5 questions: a 2 mark, a 5 mark, two 9 mark and one 12 mark question. * The 2017 paper 1 contained 5 questions: a 5 mark, three 6 mark and one 12 mark question. == Section D practice / predicted tasks == === Input validation [high confidence] === Currently the player can enter data in response to __ChoosePile() (real exam question tell you where to intervene) that will break the game. 🌶️ - VALUES - Assume the user will enter an int, but keep on asking the coordinate until it is within the bounds. 🌶️🌶️ - TYPE - the user can't be trusted to not drool on the keyboard or smth. also check it's an int. If it isn't you should display an error message - feel free to make it very sarcastic, before asking the coordinate again ==== Test ==== VALUES: Press return for default game, then Player 1, enter your choice: T Enter x coordinate: -1 Enter x coordinate: 0 Enter x coordinate: 7 Enter x coordinate: 5 Enter y coordinate: 0 Enter y coordinate: 6 Enter x coordinate: 4 Enter y coordinate: 7 Enter y coordinate: 6 TYPE: Enter filename to load or just press Enter to play the default game: MENU B. Display the board D. Display the discard T. Take turn S. Display scores Player 1, enter your choice: T Enter x coordinate: foo an integer, please Enter x coordinate: 1 Enter y coordinate: bar an integer, please Enter y coordinate: 1 Enter x coordinate: 1 Enter y coordinate: 2 Player 1, you had two matching A tiles; your score has increased by 2 === The one that explains `from math import sqrt` [high confidence] === ⚠️The fact we go in with N piles, then need to import mths sqrt may indicate a task using sqrt. Given the setup, this has got to be distance between two piles. Hence: 🌶️🌶️🌶️Change the scoring of a matching pair by adding to the tile scores the distance between the two matched piles, rounded down to the nearest integer. === Pile clear bonus [high confidence] === The game supports ''almost'' everything so that when a player match empties a pile, they get a bonus score. The one thing that's missing is in the scoring. 🌶️🌶️ Change the scoring so that if a match empties a pile, the bonus for the pile is awarded to the player. You'll need to add a getter for Pile._Bonus. ==== Test ==== Two cases, as it's possible to empty one or both piles with a match. -game1.txt tells you which piles have the same sequence, that will give you a match that should score both bonus - there's a test sequence to be found that will have one of the player score a single match. === Saving [med-high confidence] === If you've read the code closely, you'll have noticed the game1.txt is not just a description of game ''configuration'' - it is in fact a game ''save.'' If you've just run the game, you won't have noticed, because game1 is a save at the very start of the game. So they've given you a ''load'' function. Do you see what's coming over the horizon? ==== 👉Add a new menu item "S Save game". This should then call a new method on the Game class, which asks the user for a savename, and writes the game to a file, appending .txt, in the format of game1.txt - here too, you'll need a Pile.GetBonus(). ==== === Making the game both more and less boring [low, but good practice] === At the moment -certainly in the early game, the default game can be a bit boring, because every pile is the same: an A-tile, for one point a B tile for 2, another A,1. We can make the game less boring, but also longer, and long-term excitement turns out quite boring: 👉 Instead of the default game have all the same piles, keep the same proportion (that's 72 A,1, 36 B,2), but build the pile at random. Basic: If you use a random draw, the game might end up not being endable by zero tiles, but no one gets that far anyway. Fancy: actually draw 72 A1 and 36 B2, as if from a pile of tile, to randomly make the board === Party mode [low-mid confidence] === The architecture of the game, and what's visible in game1.txt means the game can support an arbitrary number of players. It is just that both the hard-coded default mode and the save file happen to have two players It's lucky, because although any player can hit the maximum score at which the game ends, the winner is always whoever between player 1 and 2 has the highest score, which is enough to run a 6-player AQA skeleton code gaming night like many Y13 enjoy. So you'll need to fix that. 🌶️🌶️🌶️ When starting a default game, it should default to a 6x6 board, and whatever basic pile o' tiles layout we're running (see 'making the game both more and less boring'), but then ask for number of players (validate for an int >0), then, for each, their names (no validation). You will need to fix the scoring bug, although testing that would require engineering a three-way game that ends with a win of the third one. === Streak === Add an attribute to the Player class storing the number of consecutive wins initalising at zero , with a getter, as well as an increment() and reset() method. Add the value of the streak to the player score when scoring, we should increment if it was a win. If the turn was not a match, call reset() the streak counter to zero. === PSYCH! === * extend the Player class to store last_move, a [(x,y) (x,y)] list of tuples, and a WasMatch boolean, initialising to False * Also getters and setters for both * On every move, set the last_move record and the boolean accordingly. * Here's the psych: if the last move was a miss, AND in the current move the player has selected one of the same piles has their last failed move, ''swap the tiles'' of the last move, before turning them over, with a 10% chance, maybe 50% to test it - you'll need to import random either way. lcbmduq096shnkz0tcq7z1shr3xon2n 2835020 2835019 2026-09-29T10:28:01Z MrBerard 3111808 Added links back in 2835020 wikitext text/x-wiki This year Skeleton Program is called "Pile O' Tiles", a two player (arguably multiplayer, see below) version of the classic match-2 memory game.</blockquote> == From AQA's preliminary material document: == <blockquote>Piles O’ Tiles is a tile matching memory game. The board consists of a two-dimensional grid containing piles of tiles. The tiles are face-down so players cannot see what is on the tiles. Each tile has a symbol (an uppercase letter) and a points value. Players take it in turn to choose two piles. The top tile of those two piles is turned over so that all players can see what is on those tiles. If the two tiles have the same symbol on them then they are removed from the piles and the points for the two tiles are added to the player’s score. The two tiles are then added to a discard. If the two tiles do not have the same symbol on them then they are turned face-down again and put back on the piles they came from. When the game ends the player with the highest score is the winner. In the Skeleton Program the board is shown as a two-dimensional grid with x and y coordinates. Each pile is shown as an integer value that denotes the number of tiles currently in the pile.</blockquote> Then a diagram showing the game's display, with a 6x6 board, rendered as a grid with the origin in the bottom left, row numbers on the left, and column numbers on the bottom<blockquote>In the Skeleton Program there is a default version of the game that uses a 6×6 grid. It is also possible to load a game from a file. The Data File game1.txt contains a game that can be loaded instead of using the default game. </blockquote>The AQA doc has the content of game1.txt, but we can do better. == Annotated game1.txt == 16 # Grid size in Nsquares - the grid is the a square of side sqrt of that 2,A,1,E,1,H,3,J,2 # Those are piles (of tiles) 0,E,1,Q,4,K,4,J,2,J,2 # arranged linearly (here, 16 of them) 0,O,1,E,1,H,3 # the first element is the bonus for the tile, 0,O,1,E,1,H,3 # then pairs of letters, score 0,A,1,E,1,H,3 0,A,1,H,1,H,3 0,E,1,E,1,H,3 0,A,1,Q,4,F,3 0,U,1,E,1,H,3 0,A,1,E,1,H,3 4,U,5,E,1,K,4 2,A,1,K,4,H,3 0,A,1,E,1,K,1 0,R,1,K,3,H,3 0,R,1,F,3,K,7 0,A,1,E,1,H,3 2 # Number of players Boris,0 # player name, score Poppy,0 # Discard, in pairs of letters, score - here it's empty 15 # max score 0 # whose turn it is (0 is player 1, Boris) 0 # turns since the the last winning move (matching tiles) by any player # Gamefiles are saves of the gamestate, but here this is the start of a game. ## From [[github:WBgrok/AQA-SKCD-2027/tree/main|https://github.com/WBgrok/AQA-SKCD-2027]] ## Courtesy of [https://www.youtube.com/@MrBerard MrBerard] - https://www.youtube.com/watch?v=4n29LNaFdsE 😎👉👉 == Annotated Skeleton Code == 👉https://github.com/WBgrok/AQA-SKCD-2027/blob/main/Annotated.py == Section D mark distribution comparison == Mark distribution for this year: '''- TK''' (please fill in if you have the EAD) These marks include the screen capture(s), so the marks for the coding will be 1-2 marks lower. * The 2026 paper 1 contained 4 questions: a 5 mark, an 8 mark and two 12 mark questions. * The 2025 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2024 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2023 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 10 mark and one 13 mark question. * The 2022 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 11 mark and one 13 mark question. * The 2021 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 9 mark and one 14 mark question. * The 2020 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 11 mark and one 12 mark question. * The 2019 paper 1 contained 4 questions: a 5 mark, an 8 mark, a 9 mark and one 13 mark question. * The 2018 paper 1 contained 5 questions: a 2 mark, a 5 mark, two 9 mark and one 12 mark question. * The 2017 paper 1 contained 5 questions: a 5 mark, three 6 mark and one 12 mark question. == Section D practice / predicted tasks == === Input validation [high confidence] === Currently the player can enter data in response to __ChoosePile() (real exam question tell you where to intervene) that will break the game. 🌶️ - VALUES - Assume the user will enter an int, but keep on asking the coordinate until it is within the bounds. 🌶️🌶️ - TYPE - the user can't be trusted to not drool on the keyboard or smth. also check it's an int. If it isn't you should display an error message - feel free to make it very sarcastic, before asking the coordinate again ==== Test ==== VALUES: Press return for default game, then Player 1, enter your choice: T Enter x coordinate: -1 Enter x coordinate: 0 Enter x coordinate: 7 Enter x coordinate: 5 Enter y coordinate: 0 Enter y coordinate: 6 Enter x coordinate: 4 Enter y coordinate: 7 Enter y coordinate: 6 TYPE: Enter filename to load or just press Enter to play the default game: MENU B. Display the board D. Display the discard T. Take turn S. Display scores Player 1, enter your choice: T Enter x coordinate: foo an integer, please Enter x coordinate: 1 Enter y coordinate: bar an integer, please Enter y coordinate: 1 Enter x coordinate: 1 Enter y coordinate: 2 Player 1, you had two matching A tiles; your score has increased by 2 === The one that explains `from math import sqrt` [high confidence] === ⚠️The fact we go in with N piles, then need to import mths sqrt may indicate a task using sqrt. Given the setup, this has got to be distance between two piles. Hence: 🌶️🌶️🌶️Change the scoring of a matching pair by adding to the tile scores the distance between the two matched piles, rounded down to the nearest integer. === Pile clear bonus [high confidence] === The game supports ''almost'' everything so that when a player match empties a pile, they get a bonus score. The one thing that's missing is in the scoring. 🌶️🌶️ Change the scoring so that if a match empties a pile, the bonus for the pile is awarded to the player. You'll need to add a getter for Pile._Bonus. ==== Test ==== Two cases, as it's possible to empty one or both piles with a match. -game1.txt tells you which piles have the same sequence, that will give you a match that should score both bonus - there's a test sequence to be found that will have one of the player score a single match. === Saving [med-high confidence] === If you've read the code closely, you'll have noticed the game1.txt is not just a description of game ''configuration'' - it is in fact a game ''save.'' If you've just run the game, you won't have noticed, because game1 is a save at the very start of the game. So they've given you a ''load'' function. Do you see what's coming over the horizon? ==== 👉Add a new menu item "S Save game". This should then call a new method on the Game class, which asks the user for a savename, and writes the game to a file, appending .txt, in the format of game1.txt - here too, you'll need a Pile.GetBonus(). ==== === Making the game both more and less boring [low, but good practice] === At the moment -certainly in the early game, the default game can be a bit boring, because every pile is the same: an A-tile, for one point a B tile for 2, another A,1. We can make the game less boring, but also longer, and long-term excitement turns out quite boring: 👉 Instead of the default game have all the same piles, keep the same proportion (that's 72 A,1, 36 B,2), but build the pile at random. Basic: If you use a random draw, the game might end up not being endable by zero tiles, but no one gets that far anyway. Fancy: actually draw 72 A1 and 36 B2, as if from a pile of tile, to randomly make the board === Party mode [low-mid confidence] === The architecture of the game, and what's visible in game1.txt means the game can support an arbitrary number of players. It is just that both the hard-coded default mode and the save file happen to have two players It's lucky, because although any player can hit the maximum score at which the game ends, the winner is always whoever between player 1 and 2 has the highest score, which is enough to run a 6-player AQA skeleton code gaming night like many Y13 enjoy. So you'll need to fix that. 🌶️🌶️🌶️ When starting a default game, it should default to a 6x6 board, and whatever basic pile o' tiles layout we're running (see 'making the game both more and less boring'), but then ask for number of players (validate for an int >0), then, for each, their names (no validation). You will need to fix the scoring bug, although testing that would require engineering a three-way game that ends with a win of the third one. === Streak === Add an attribute to the Player class storing the number of consecutive wins initalising at zero , with a getter, as well as an increment() and reset() method. Add the value of the streak to the player score when scoring, we should increment if it was a win. If the turn was not a match, call reset() the streak counter to zero. === PSYCH! === * extend the Player class to store last_move, a [(x,y) (x,y)] list of tuples, and a WasMatch boolean, initialising to False * Also getters and setters for both * On every move, set the last_move record and the boolean accordingly. * Here's the psych: if the last move was a miss, AND in the current move the player has selected one of the same piles has their last failed move, ''swap the tiles'' of the last move, before turning them over, with a 10% chance, maybe 50% to test it - you'll need to import random either way. rm0mpej6ujn18ynzxszalx98lullclg 2835025 2835020 2026-09-29T10:40:55Z MrBerard 3111808 /* The one that explains `from math import sqrt` [high confidence] */ added test case 2835025 wikitext text/x-wiki This year Skeleton Program is called "Pile O' Tiles", a two player (arguably multiplayer, see below) version of the classic match-2 memory game.</blockquote> == From AQA's preliminary material document: == <blockquote>Piles O’ Tiles is a tile matching memory game. The board consists of a two-dimensional grid containing piles of tiles. The tiles are face-down so players cannot see what is on the tiles. Each tile has a symbol (an uppercase letter) and a points value. Players take it in turn to choose two piles. The top tile of those two piles is turned over so that all players can see what is on those tiles. If the two tiles have the same symbol on them then they are removed from the piles and the points for the two tiles are added to the player’s score. The two tiles are then added to a discard. If the two tiles do not have the same symbol on them then they are turned face-down again and put back on the piles they came from. When the game ends the player with the highest score is the winner. In the Skeleton Program the board is shown as a two-dimensional grid with x and y coordinates. Each pile is shown as an integer value that denotes the number of tiles currently in the pile.</blockquote> Then a diagram showing the game's display, with a 6x6 board, rendered as a grid with the origin in the bottom left, row numbers on the left, and column numbers on the bottom<blockquote>In the Skeleton Program there is a default version of the game that uses a 6×6 grid. It is also possible to load a game from a file. The Data File game1.txt contains a game that can be loaded instead of using the default game. </blockquote>The AQA doc has the content of game1.txt, but we can do better. == Annotated game1.txt == 16 # Grid size in Nsquares - the grid is the a square of side sqrt of that 2,A,1,E,1,H,3,J,2 # Those are piles (of tiles) 0,E,1,Q,4,K,4,J,2,J,2 # arranged linearly (here, 16 of them) 0,O,1,E,1,H,3 # the first element is the bonus for the tile, 0,O,1,E,1,H,3 # then pairs of letters, score 0,A,1,E,1,H,3 0,A,1,H,1,H,3 0,E,1,E,1,H,3 0,A,1,Q,4,F,3 0,U,1,E,1,H,3 0,A,1,E,1,H,3 4,U,5,E,1,K,4 2,A,1,K,4,H,3 0,A,1,E,1,K,1 0,R,1,K,3,H,3 0,R,1,F,3,K,7 0,A,1,E,1,H,3 2 # Number of players Boris,0 # player name, score Poppy,0 # Discard, in pairs of letters, score - here it's empty 15 # max score 0 # whose turn it is (0 is player 1, Boris) 0 # turns since the the last winning move (matching tiles) by any player # Gamefiles are saves of the gamestate, but here this is the start of a game. ## From [[github:WBgrok/AQA-SKCD-2027/tree/main|https://github.com/WBgrok/AQA-SKCD-2027]] ## Courtesy of [https://www.youtube.com/@MrBerard MrBerard] - https://www.youtube.com/watch?v=4n29LNaFdsE 😎👉👉 == Annotated Skeleton Code == 👉https://github.com/WBgrok/AQA-SKCD-2027/blob/main/Annotated.py == Section D mark distribution comparison == Mark distribution for this year: '''- TK''' (please fill in if you have the EAD) These marks include the screen capture(s), so the marks for the coding will be 1-2 marks lower. * The 2026 paper 1 contained 4 questions: a 5 mark, an 8 mark and two 12 mark questions. * The 2025 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2024 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2023 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 10 mark and one 13 mark question. * The 2022 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 11 mark and one 13 mark question. * The 2021 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 9 mark and one 14 mark question. * The 2020 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 11 mark and one 12 mark question. * The 2019 paper 1 contained 4 questions: a 5 mark, an 8 mark, a 9 mark and one 13 mark question. * The 2018 paper 1 contained 5 questions: a 2 mark, a 5 mark, two 9 mark and one 12 mark question. * The 2017 paper 1 contained 5 questions: a 5 mark, three 6 mark and one 12 mark question. == Section D practice / predicted tasks == === Input validation [high confidence] === Currently the player can enter data in response to __ChoosePile() (real exam question tell you where to intervene) that will break the game. 🌶️ - VALUES - Assume the user will enter an int, but keep on asking the coordinate until it is within the bounds. 🌶️🌶️ - TYPE - the user can't be trusted to not drool on the keyboard or smth. also check it's an int. If it isn't you should display an error message - feel free to make it very sarcastic, before asking the coordinate again ==== Test ==== VALUES: Press return for default game, then Player 1, enter your choice: T Enter x coordinate: -1 Enter x coordinate: 0 Enter x coordinate: 7 Enter x coordinate: 5 Enter y coordinate: 0 Enter y coordinate: 6 Enter x coordinate: 4 Enter y coordinate: 7 Enter y coordinate: 6 TYPE: Enter filename to load or just press Enter to play the default game: MENU B. Display the board D. Display the discard T. Take turn S. Display scores Player 1, enter your choice: T Enter x coordinate: foo an integer, please Enter x coordinate: 1 Enter y coordinate: bar an integer, please Enter y coordinate: 1 Enter x coordinate: 1 Enter y coordinate: 2 Player 1, you had two matching A tiles; your score has increased by 2 === The one that explains `from math import sqrt` [high confidence] === ⚠️The fact we go in with N piles, then need to import mths sqrt may indicate a task using sqrt. Given the setup, this has got to be distance between two piles. Hence: 🌶️🌶️🌶️Change the scoring of a matching pair by adding to the tile scores the distance between the two matched piles, rounded down to the nearest integer. '''TESTING''' * [Return] for default game * T * 1 * 1 * 4 * 4 * Player 1 should score 5: 2 for the tiles, 3 for the distance === Pile clear bonus [high confidence] === The game supports ''almost'' everything so that when a player match empties a pile, they get a bonus score. The one thing that's missing is in the scoring. 🌶️🌶️ Change the scoring so that if a match empties a pile, the bonus for the pile is awarded to the player. You'll need to add a getter for Pile._Bonus. ==== Test ==== Two cases, as it's possible to empty one or both piles with a match. -game1.txt tells you which piles have the same sequence, that will give you a match that should score both bonus - there's a test sequence to be found that will have one of the player score a single match. === Saving [med-high confidence] === If you've read the code closely, you'll have noticed the game1.txt is not just a description of game ''configuration'' - it is in fact a game ''save.'' If you've just run the game, you won't have noticed, because game1 is a save at the very start of the game. So they've given you a ''load'' function. Do you see what's coming over the horizon? ==== 👉Add a new menu item "S Save game". This should then call a new method on the Game class, which asks the user for a savename, and writes the game to a file, appending .txt, in the format of game1.txt - here too, you'll need a Pile.GetBonus(). ==== === Making the game both more and less boring [low, but good practice] === At the moment -certainly in the early game, the default game can be a bit boring, because every pile is the same: an A-tile, for one point a B tile for 2, another A,1. We can make the game less boring, but also longer, and long-term excitement turns out quite boring: 👉 Instead of the default game have all the same piles, keep the same proportion (that's 72 A,1, 36 B,2), but build the pile at random. Basic: If you use a random draw, the game might end up not being endable by zero tiles, but no one gets that far anyway. Fancy: actually draw 72 A1 and 36 B2, as if from a pile of tile, to randomly make the board === Party mode [low-mid confidence] === The architecture of the game, and what's visible in game1.txt means the game can support an arbitrary number of players. It is just that both the hard-coded default mode and the save file happen to have two players It's lucky, because although any player can hit the maximum score at which the game ends, the winner is always whoever between player 1 and 2 has the highest score, which is enough to run a 6-player AQA skeleton code gaming night like many Y13 enjoy. So you'll need to fix that. 🌶️🌶️🌶️ When starting a default game, it should default to a 6x6 board, and whatever basic pile o' tiles layout we're running (see 'making the game both more and less boring'), but then ask for number of players (validate for an int >0), then, for each, their names (no validation). You will need to fix the scoring bug, although testing that would require engineering a three-way game that ends with a win of the third one. === Streak === Add an attribute to the Player class storing the number of consecutive wins initalising at zero , with a getter, as well as an increment() and reset() method. Add the value of the streak to the player score when scoring, we should increment if it was a win. If the turn was not a match, call reset() the streak counter to zero. === PSYCH! === * extend the Player class to store last_move, a [(x,y) (x,y)] list of tuples, and a WasMatch boolean, initialising to False * Also getters and setters for both * On every move, set the last_move record and the boolean accordingly. * Here's the psych: if the last move was a miss, AND in the current move the player has selected one of the same piles has their last failed move, ''swap the tiles'' of the last move, before turning them over, with a 10% chance, maybe 50% to test it - you'll need to import random either way. ao0fna32v27w5ki6c3crfv5j05fzxrf 2835028 2835025 2026-09-29T10:48:28Z MrBerard 3111808 /* The one that explains `from math import sqrt` [high confidence] */ Test case 2835028 wikitext text/x-wiki This year Skeleton Program is called "Pile O' Tiles", a two player (arguably multiplayer, see below) version of the classic match-2 memory game.</blockquote> == From AQA's preliminary material document: == <blockquote>Piles O’ Tiles is a tile matching memory game. The board consists of a two-dimensional grid containing piles of tiles. The tiles are face-down so players cannot see what is on the tiles. Each tile has a symbol (an uppercase letter) and a points value. Players take it in turn to choose two piles. The top tile of those two piles is turned over so that all players can see what is on those tiles. If the two tiles have the same symbol on them then they are removed from the piles and the points for the two tiles are added to the player’s score. The two tiles are then added to a discard. If the two tiles do not have the same symbol on them then they are turned face-down again and put back on the piles they came from. When the game ends the player with the highest score is the winner. In the Skeleton Program the board is shown as a two-dimensional grid with x and y coordinates. Each pile is shown as an integer value that denotes the number of tiles currently in the pile.</blockquote> Then a diagram showing the game's display, with a 6x6 board, rendered as a grid with the origin in the bottom left, row numbers on the left, and column numbers on the bottom<blockquote>In the Skeleton Program there is a default version of the game that uses a 6×6 grid. It is also possible to load a game from a file. The Data File game1.txt contains a game that can be loaded instead of using the default game. </blockquote>The AQA doc has the content of game1.txt, but we can do better. == Annotated game1.txt == 16 # Grid size in Nsquares - the grid is the a square of side sqrt of that 2,A,1,E,1,H,3,J,2 # Those are piles (of tiles) 0,E,1,Q,4,K,4,J,2,J,2 # arranged linearly (here, 16 of them) 0,O,1,E,1,H,3 # the first element is the bonus for the tile, 0,O,1,E,1,H,3 # then pairs of letters, score 0,A,1,E,1,H,3 0,A,1,H,1,H,3 0,E,1,E,1,H,3 0,A,1,Q,4,F,3 0,U,1,E,1,H,3 0,A,1,E,1,H,3 4,U,5,E,1,K,4 2,A,1,K,4,H,3 0,A,1,E,1,K,1 0,R,1,K,3,H,3 0,R,1,F,3,K,7 0,A,1,E,1,H,3 2 # Number of players Boris,0 # player name, score Poppy,0 # Discard, in pairs of letters, score - here it's empty 15 # max score 0 # whose turn it is (0 is player 1, Boris) 0 # turns since the the last winning move (matching tiles) by any player # Gamefiles are saves of the gamestate, but here this is the start of a game. ## From [[github:WBgrok/AQA-SKCD-2027/tree/main|https://github.com/WBgrok/AQA-SKCD-2027]] ## Courtesy of [https://www.youtube.com/@MrBerard MrBerard] - https://www.youtube.com/watch?v=4n29LNaFdsE 😎👉👉 == Annotated Skeleton Code == 👉https://github.com/WBgrok/AQA-SKCD-2027/blob/main/Annotated.py == Section D mark distribution comparison == Mark distribution for this year: '''- TK''' (please fill in if you have the EAD) These marks include the screen capture(s), so the marks for the coding will be 1-2 marks lower. * The 2026 paper 1 contained 4 questions: a 5 mark, an 8 mark and two 12 mark questions. * The 2025 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2024 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2023 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 10 mark and one 13 mark question. * The 2022 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 11 mark and one 13 mark question. * The 2021 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 9 mark and one 14 mark question. * The 2020 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 11 mark and one 12 mark question. * The 2019 paper 1 contained 4 questions: a 5 mark, an 8 mark, a 9 mark and one 13 mark question. * The 2018 paper 1 contained 5 questions: a 2 mark, a 5 mark, two 9 mark and one 12 mark question. * The 2017 paper 1 contained 5 questions: a 5 mark, three 6 mark and one 12 mark question. == Section D practice / predicted tasks == === Input validation [high confidence] === Currently the player can enter data in response to __ChoosePile() (real exam question tell you where to intervene) that will break the game. 🌶️ - VALUES - Assume the user will enter an int, but keep on asking the coordinate until it is within the bounds. 🌶️🌶️ - TYPE - the user can't be trusted to not drool on the keyboard or smth. also check it's an int. If it isn't you should display an error message - feel free to make it very sarcastic, before asking the coordinate again ==== Test ==== VALUES: Press return for default game, then Player 1, enter your choice: T Enter x coordinate: -1 Enter x coordinate: 0 Enter x coordinate: 7 Enter x coordinate: 5 Enter y coordinate: 0 Enter y coordinate: 6 Enter x coordinate: 4 Enter y coordinate: 7 Enter y coordinate: 6 TYPE: Enter filename to load or just press Enter to play the default game: MENU B. Display the board D. Display the discard T. Take turn S. Display scores Player 1, enter your choice: T Enter x coordinate: foo an integer, please Enter x coordinate: 1 Enter y coordinate: bar an integer, please Enter y coordinate: 1 Enter x coordinate: 1 Enter y coordinate: 2 Player 1, you had two matching A tiles; your score has increased by 2 === The one that explains `from math import sqrt` [high confidence] === ⚠️The fact we go in with N piles, then need to import mths sqrt may indicate a task using sqrt. Given the setup, this has got to be distance between two piles. Hence: 🌶️🌶️🌶️Change the scoring of a matching pair by adding to the tile scores the distance between the two matched piles, rounded down to the nearest integer. '''TESTING''' * [Return] for default game * T * 1 * 1 * 4 * 4 * Player 1 should score 5: 2 for the tiles, 4 for the distance (sqrt(9+9)) === Pile clear bonus [high confidence] === The game supports ''almost'' everything so that when a player match empties a pile, they get a bonus score. The one thing that's missing is in the scoring. 🌶️🌶️ Change the scoring so that if a match empties a pile, the bonus for the pile is awarded to the player. You'll need to add a getter for Pile._Bonus. ==== Test ==== Two cases, as it's possible to empty one or both piles with a match. -game1.txt tells you which piles have the same sequence, that will give you a match that should score both bonus - there's a test sequence to be found that will have one of the player score a single match. === Saving [med-high confidence] === If you've read the code closely, you'll have noticed the game1.txt is not just a description of game ''configuration'' - it is in fact a game ''save.'' If you've just run the game, you won't have noticed, because game1 is a save at the very start of the game. So they've given you a ''load'' function. Do you see what's coming over the horizon? ==== 👉Add a new menu item "S Save game". This should then call a new method on the Game class, which asks the user for a savename, and writes the game to a file, appending .txt, in the format of game1.txt - here too, you'll need a Pile.GetBonus(). ==== === Making the game both more and less boring [low, but good practice] === At the moment -certainly in the early game, the default game can be a bit boring, because every pile is the same: an A-tile, for one point a B tile for 2, another A,1. We can make the game less boring, but also longer, and long-term excitement turns out quite boring: 👉 Instead of the default game have all the same piles, keep the same proportion (that's 72 A,1, 36 B,2), but build the pile at random. Basic: If you use a random draw, the game might end up not being endable by zero tiles, but no one gets that far anyway. Fancy: actually draw 72 A1 and 36 B2, as if from a pile of tile, to randomly make the board === Party mode [low-mid confidence] === The architecture of the game, and what's visible in game1.txt means the game can support an arbitrary number of players. It is just that both the hard-coded default mode and the save file happen to have two players It's lucky, because although any player can hit the maximum score at which the game ends, the winner is always whoever between player 1 and 2 has the highest score, which is enough to run a 6-player AQA skeleton code gaming night like many Y13 enjoy. So you'll need to fix that. 🌶️🌶️🌶️ When starting a default game, it should default to a 6x6 board, and whatever basic pile o' tiles layout we're running (see 'making the game both more and less boring'), but then ask for number of players (validate for an int >0), then, for each, their names (no validation). You will need to fix the scoring bug, although testing that would require engineering a three-way game that ends with a win of the third one. === Streak === Add an attribute to the Player class storing the number of consecutive wins initalising at zero , with a getter, as well as an increment() and reset() method. Add the value of the streak to the player score when scoring, we should increment if it was a win. If the turn was not a match, call reset() the streak counter to zero. === PSYCH! === * extend the Player class to store last_move, a [(x,y) (x,y)] list of tuples, and a WasMatch boolean, initialising to False * Also getters and setters for both * On every move, set the last_move record and the boolean accordingly. * Here's the psych: if the last move was a miss, AND in the current move the player has selected one of the same piles has their last failed move, ''swap the tiles'' of the last move, before turning them over, with a 10% chance, maybe 50% to test it - you'll need to import random either way. tn722aanq30323z8fzlw2tx5jat4bwk 2835029 2835028 2026-09-29T10:51:15Z MrBerard 3111808 /* The one that explains `from math import sqrt` [high confidence] */ Tests 2835029 wikitext text/x-wiki This year Skeleton Program is called "Pile O' Tiles", a two player (arguably multiplayer, see below) version of the classic match-2 memory game.</blockquote> == From AQA's preliminary material document: == <blockquote>Piles O’ Tiles is a tile matching memory game. The board consists of a two-dimensional grid containing piles of tiles. The tiles are face-down so players cannot see what is on the tiles. Each tile has a symbol (an uppercase letter) and a points value. Players take it in turn to choose two piles. The top tile of those two piles is turned over so that all players can see what is on those tiles. If the two tiles have the same symbol on them then they are removed from the piles and the points for the two tiles are added to the player’s score. The two tiles are then added to a discard. If the two tiles do not have the same symbol on them then they are turned face-down again and put back on the piles they came from. When the game ends the player with the highest score is the winner. In the Skeleton Program the board is shown as a two-dimensional grid with x and y coordinates. Each pile is shown as an integer value that denotes the number of tiles currently in the pile.</blockquote> Then a diagram showing the game's display, with a 6x6 board, rendered as a grid with the origin in the bottom left, row numbers on the left, and column numbers on the bottom<blockquote>In the Skeleton Program there is a default version of the game that uses a 6×6 grid. It is also possible to load a game from a file. The Data File game1.txt contains a game that can be loaded instead of using the default game. </blockquote>The AQA doc has the content of game1.txt, but we can do better. == Annotated game1.txt == 16 # Grid size in Nsquares - the grid is the a square of side sqrt of that 2,A,1,E,1,H,3,J,2 # Those are piles (of tiles) 0,E,1,Q,4,K,4,J,2,J,2 # arranged linearly (here, 16 of them) 0,O,1,E,1,H,3 # the first element is the bonus for the tile, 0,O,1,E,1,H,3 # then pairs of letters, score 0,A,1,E,1,H,3 0,A,1,H,1,H,3 0,E,1,E,1,H,3 0,A,1,Q,4,F,3 0,U,1,E,1,H,3 0,A,1,E,1,H,3 4,U,5,E,1,K,4 2,A,1,K,4,H,3 0,A,1,E,1,K,1 0,R,1,K,3,H,3 0,R,1,F,3,K,7 0,A,1,E,1,H,3 2 # Number of players Boris,0 # player name, score Poppy,0 # Discard, in pairs of letters, score - here it's empty 15 # max score 0 # whose turn it is (0 is player 1, Boris) 0 # turns since the the last winning move (matching tiles) by any player # Gamefiles are saves of the gamestate, but here this is the start of a game. ## From [[github:WBgrok/AQA-SKCD-2027/tree/main|https://github.com/WBgrok/AQA-SKCD-2027]] ## Courtesy of [https://www.youtube.com/@MrBerard MrBerard] - https://www.youtube.com/watch?v=4n29LNaFdsE 😎👉👉 == Annotated Skeleton Code == 👉https://github.com/WBgrok/AQA-SKCD-2027/blob/main/Annotated.py == Section D mark distribution comparison == Mark distribution for this year: '''- TK''' (please fill in if you have the EAD) These marks include the screen capture(s), so the marks for the coding will be 1-2 marks lower. * The 2026 paper 1 contained 4 questions: a 5 mark, an 8 mark and two 12 mark questions. * The 2025 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2024 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2023 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 10 mark and one 13 mark question. * The 2022 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 11 mark and one 13 mark question. * The 2021 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 9 mark and one 14 mark question. * The 2020 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 11 mark and one 12 mark question. * The 2019 paper 1 contained 4 questions: a 5 mark, an 8 mark, a 9 mark and one 13 mark question. * The 2018 paper 1 contained 5 questions: a 2 mark, a 5 mark, two 9 mark and one 12 mark question. * The 2017 paper 1 contained 5 questions: a 5 mark, three 6 mark and one 12 mark question. == Section D practice / predicted tasks == === Input validation [high confidence] === Currently the player can enter data in response to __ChoosePile() (real exam question tell you where to intervene) that will break the game. 🌶️ - VALUES - Assume the user will enter an int, but keep on asking the coordinate until it is within the bounds. 🌶️🌶️ - TYPE - the user can't be trusted to not drool on the keyboard or smth. also check it's an int. If it isn't you should display an error message - feel free to make it very sarcastic, before asking the coordinate again ==== Test ==== VALUES: Press return for default game, then Player 1, enter your choice: T Enter x coordinate: -1 Enter x coordinate: 0 Enter x coordinate: 7 Enter x coordinate: 5 Enter y coordinate: 0 Enter y coordinate: 6 Enter x coordinate: 4 Enter y coordinate: 7 Enter y coordinate: 6 TYPE: Enter filename to load or just press Enter to play the default game: MENU B. Display the board D. Display the discard T. Take turn S. Display scores Player 1, enter your choice: T Enter x coordinate: foo an integer, please Enter x coordinate: 1 Enter y coordinate: bar an integer, please Enter y coordinate: 1 Enter x coordinate: 1 Enter y coordinate: 2 Player 1, you had two matching A tiles; your score has increased by 2 === The one that explains `from math import sqrt` [high confidence] === ⚠️The fact we go in with N piles, then need to import mths sqrt may indicate a task using sqrt. Given the setup, this has got to be distance between two piles. Hence: 🌶️🌶️🌶️Change the scoring of a matching pair by adding to the tile scores the distance between the two matched piles, rounded down to the nearest integer. '''TESTING''' * [Return] for default game * T * 1 * 1 * 4 * 4 * Player 1 should score 6: 2 for the tiles, 4 for the distance (sqrt(9+9)) * T * 2 * 1 * 2 * 2 * Player 2 should score 5: 3 for the tiles, 1 for the distance (vertical only) === Pile clear bonus [high confidence] === The game supports ''almost'' everything so that when a player match empties a pile, they get a bonus score. The one thing that's missing is in the scoring. 🌶️🌶️ Change the scoring so that if a match empties a pile, the bonus for the pile is awarded to the player. You'll need to add a getter for Pile._Bonus. ==== Test ==== Two cases, as it's possible to empty one or both piles with a match. -game1.txt tells you which piles have the same sequence, that will give you a match that should score both bonus - there's a test sequence to be found that will have one of the player score a single match. === Saving [med-high confidence] === If you've read the code closely, you'll have noticed the game1.txt is not just a description of game ''configuration'' - it is in fact a game ''save.'' If you've just run the game, you won't have noticed, because game1 is a save at the very start of the game. So they've given you a ''load'' function. Do you see what's coming over the horizon? ==== 👉Add a new menu item "S Save game". This should then call a new method on the Game class, which asks the user for a savename, and writes the game to a file, appending .txt, in the format of game1.txt - here too, you'll need a Pile.GetBonus(). ==== === Making the game both more and less boring [low, but good practice] === At the moment -certainly in the early game, the default game can be a bit boring, because every pile is the same: an A-tile, for one point a B tile for 2, another A,1. We can make the game less boring, but also longer, and long-term excitement turns out quite boring: 👉 Instead of the default game have all the same piles, keep the same proportion (that's 72 A,1, 36 B,2), but build the pile at random. Basic: If you use a random draw, the game might end up not being endable by zero tiles, but no one gets that far anyway. Fancy: actually draw 72 A1 and 36 B2, as if from a pile of tile, to randomly make the board === Party mode [low-mid confidence] === The architecture of the game, and what's visible in game1.txt means the game can support an arbitrary number of players. It is just that both the hard-coded default mode and the save file happen to have two players It's lucky, because although any player can hit the maximum score at which the game ends, the winner is always whoever between player 1 and 2 has the highest score, which is enough to run a 6-player AQA skeleton code gaming night like many Y13 enjoy. So you'll need to fix that. 🌶️🌶️🌶️ When starting a default game, it should default to a 6x6 board, and whatever basic pile o' tiles layout we're running (see 'making the game both more and less boring'), but then ask for number of players (validate for an int >0), then, for each, their names (no validation). You will need to fix the scoring bug, although testing that would require engineering a three-way game that ends with a win of the third one. === Streak === Add an attribute to the Player class storing the number of consecutive wins initalising at zero , with a getter, as well as an increment() and reset() method. Add the value of the streak to the player score when scoring, we should increment if it was a win. If the turn was not a match, call reset() the streak counter to zero. === PSYCH! === * extend the Player class to store last_move, a [(x,y) (x,y)] list of tuples, and a WasMatch boolean, initialising to False * Also getters and setters for both * On every move, set the last_move record and the boolean accordingly. * Here's the psych: if the last move was a miss, AND in the current move the player has selected one of the same piles has their last failed move, ''swap the tiles'' of the last move, before turning them over, with a 10% chance, maybe 50% to test it - you'll need to import random either way. bxphja1saowpnpbiebrft5khpb5csqu 2835032 2835029 2026-09-29T10:54:12Z MrBerard 3111808 /* The one that explains `from math import sqrt` [high confidence] */ fix the spec 2835032 wikitext text/x-wiki This year Skeleton Program is called "Pile O' Tiles", a two player (arguably multiplayer, see below) version of the classic match-2 memory game.</blockquote> == From AQA's preliminary material document: == <blockquote>Piles O’ Tiles is a tile matching memory game. The board consists of a two-dimensional grid containing piles of tiles. The tiles are face-down so players cannot see what is on the tiles. Each tile has a symbol (an uppercase letter) and a points value. Players take it in turn to choose two piles. The top tile of those two piles is turned over so that all players can see what is on those tiles. If the two tiles have the same symbol on them then they are removed from the piles and the points for the two tiles are added to the player’s score. The two tiles are then added to a discard. If the two tiles do not have the same symbol on them then they are turned face-down again and put back on the piles they came from. When the game ends the player with the highest score is the winner. In the Skeleton Program the board is shown as a two-dimensional grid with x and y coordinates. Each pile is shown as an integer value that denotes the number of tiles currently in the pile.</blockquote> Then a diagram showing the game's display, with a 6x6 board, rendered as a grid with the origin in the bottom left, row numbers on the left, and column numbers on the bottom<blockquote>In the Skeleton Program there is a default version of the game that uses a 6×6 grid. It is also possible to load a game from a file. The Data File game1.txt contains a game that can be loaded instead of using the default game. </blockquote>The AQA doc has the content of game1.txt, but we can do better. == Annotated game1.txt == 16 # Grid size in Nsquares - the grid is the a square of side sqrt of that 2,A,1,E,1,H,3,J,2 # Those are piles (of tiles) 0,E,1,Q,4,K,4,J,2,J,2 # arranged linearly (here, 16 of them) 0,O,1,E,1,H,3 # the first element is the bonus for the tile, 0,O,1,E,1,H,3 # then pairs of letters, score 0,A,1,E,1,H,3 0,A,1,H,1,H,3 0,E,1,E,1,H,3 0,A,1,Q,4,F,3 0,U,1,E,1,H,3 0,A,1,E,1,H,3 4,U,5,E,1,K,4 2,A,1,K,4,H,3 0,A,1,E,1,K,1 0,R,1,K,3,H,3 0,R,1,F,3,K,7 0,A,1,E,1,H,3 2 # Number of players Boris,0 # player name, score Poppy,0 # Discard, in pairs of letters, score - here it's empty 15 # max score 0 # whose turn it is (0 is player 1, Boris) 0 # turns since the the last winning move (matching tiles) by any player # Gamefiles are saves of the gamestate, but here this is the start of a game. ## From [[github:WBgrok/AQA-SKCD-2027/tree/main|https://github.com/WBgrok/AQA-SKCD-2027]] ## Courtesy of [https://www.youtube.com/@MrBerard MrBerard] - https://www.youtube.com/watch?v=4n29LNaFdsE 😎👉👉 == Annotated Skeleton Code == 👉https://github.com/WBgrok/AQA-SKCD-2027/blob/main/Annotated.py == Section D mark distribution comparison == Mark distribution for this year: '''- TK''' (please fill in if you have the EAD) These marks include the screen capture(s), so the marks for the coding will be 1-2 marks lower. * The 2026 paper 1 contained 4 questions: a 5 mark, an 8 mark and two 12 mark questions. * The 2025 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2024 paper 1 contained 4 questions: a 5 mark, a 6 mark and two 14 mark questions. * The 2023 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 10 mark and one 13 mark question. * The 2022 paper 1 contained 4 questions: a 5 mark, a 9 mark, a 11 mark and one 13 mark question. * The 2021 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 9 mark and one 14 mark question. * The 2020 paper 1 contained 4 questions: a 6 mark, an 8 mark, a 11 mark and one 12 mark question. * The 2019 paper 1 contained 4 questions: a 5 mark, an 8 mark, a 9 mark and one 13 mark question. * The 2018 paper 1 contained 5 questions: a 2 mark, a 5 mark, two 9 mark and one 12 mark question. * The 2017 paper 1 contained 5 questions: a 5 mark, three 6 mark and one 12 mark question. == Section D practice / predicted tasks == === Input validation [high confidence] === Currently the player can enter data in response to __ChoosePile() (real exam question tell you where to intervene) that will break the game. 🌶️ - VALUES - Assume the user will enter an int, but keep on asking the coordinate until it is within the bounds. 🌶️🌶️ - TYPE - the user can't be trusted to not drool on the keyboard or smth. also check it's an int. If it isn't you should display an error message - feel free to make it very sarcastic, before asking the coordinate again ==== Test ==== VALUES: Press return for default game, then Player 1, enter your choice: T Enter x coordinate: -1 Enter x coordinate: 0 Enter x coordinate: 7 Enter x coordinate: 5 Enter y coordinate: 0 Enter y coordinate: 6 Enter x coordinate: 4 Enter y coordinate: 7 Enter y coordinate: 6 TYPE: Enter filename to load or just press Enter to play the default game: MENU B. Display the board D. Display the discard T. Take turn S. Display scores Player 1, enter your choice: T Enter x coordinate: foo an integer, please Enter x coordinate: 1 Enter y coordinate: bar an integer, please Enter y coordinate: 1 Enter x coordinate: 1 Enter y coordinate: 2 Player 1, you had two matching A tiles; your score has increased by 2 === The one that explains `from math import sqrt` [high confidence] === ⚠️The fact we go in with N piles, then need to import mths sqrt may indicate a task using sqrt. Given the setup, this has got to be distance between two piles. Hence: 🌶️🌶️🌶️Change the scoring of a matching pair by adding to the tile scores the distance between the two matched piles, rounded to the nearest integer. '''TESTING''' * [Return] for default game * T * 1 * 1 * 4 * 4 * Player 1 should score 6: 2 for the tiles, 4 for the distance (sqrt(9+9)) * T * 2 * 1 * 2 * 2 * Player 2 should score 5: 3 for the tiles, 1 for the distance (vertical only) === Pile clear bonus [high confidence] === The game supports ''almost'' everything so that when a player match empties a pile, they get a bonus score. The one thing that's missing is in the scoring. 🌶️🌶️ Change the scoring so that if a match empties a pile, the bonus for the pile is awarded to the player. You'll need to add a getter for Pile._Bonus. ==== Test ==== Two cases, as it's possible to empty one or both piles with a match. -game1.txt tells you which piles have the same sequence, that will give you a match that should score both bonus - there's a test sequence to be found that will have one of the player score a single match. === Saving [med-high confidence] === If you've read the code closely, you'll have noticed the game1.txt is not just a description of game ''configuration'' - it is in fact a game ''save.'' If you've just run the game, you won't have noticed, because game1 is a save at the very start of the game. So they've given you a ''load'' function. Do you see what's coming over the horizon? ==== 👉Add a new menu item "S Save game". This should then call a new method on the Game class, which asks the user for a savename, and writes the game to a file, appending .txt, in the format of game1.txt - here too, you'll need a Pile.GetBonus(). ==== === Making the game both more and less boring [low, but good practice] === At the moment -certainly in the early game, the default game can be a bit boring, because every pile is the same: an A-tile, for one point a B tile for 2, another A,1. We can make the game less boring, but also longer, and long-term excitement turns out quite boring: 👉 Instead of the default game have all the same piles, keep the same proportion (that's 72 A,1, 36 B,2), but build the pile at random. Basic: If you use a random draw, the game might end up not being endable by zero tiles, but no one gets that far anyway. Fancy: actually draw 72 A1 and 36 B2, as if from a pile of tile, to randomly make the board === Party mode [low-mid confidence] === The architecture of the game, and what's visible in game1.txt means the game can support an arbitrary number of players. It is just that both the hard-coded default mode and the save file happen to have two players It's lucky, because although any player can hit the maximum score at which the game ends, the winner is always whoever between player 1 and 2 has the highest score, which is enough to run a 6-player AQA skeleton code gaming night like many Y13 enjoy. So you'll need to fix that. 🌶️🌶️🌶️ When starting a default game, it should default to a 6x6 board, and whatever basic pile o' tiles layout we're running (see 'making the game both more and less boring'), but then ask for number of players (validate for an int >0), then, for each, their names (no validation). You will need to fix the scoring bug, although testing that would require engineering a three-way game that ends with a win of the third one. === Streak === Add an attribute to the Player class storing the number of consecutive wins initalising at zero , with a getter, as well as an increment() and reset() method. Add the value of the streak to the player score when scoring, we should increment if it was a win. If the turn was not a match, call reset() the streak counter to zero. === PSYCH! === * extend the Player class to store last_move, a [(x,y) (x,y)] list of tuples, and a WasMatch boolean, initialising to False * Also getters and setters for both * On every move, set the last_move record and the boolean accordingly. * Here's the psych: if the last move was a miss, AND in the current move the player has selected one of the same piles has their last failed move, ''swap the tiles'' of the last move, before turning them over, with a 10% chance, maybe 50% to test it - you'll need to import random either way. k72xwmoyl0lyt15wpjo72p8vngjxvvl