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id="toc-Geometry-sublist" class="vector-toc-list"> <li id="toc-Hypersphere" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hypersphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Hypersphere</span> </div> </a> <ul id="toc-Hypersphere-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Four-dimensional_perception_in_humans" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Four-dimensional_perception_in_humans"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Four-dimensional perception in humans</span> </div> </a> <ul id="toc-Four-dimensional_perception_in_humans-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensional_analogy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dimensional_analogy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Dimensional analogy</span> </div> </a> <button aria-controls="toc-Dimensional_analogy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dimensional analogy subsection</span> </button> <ul id="toc-Dimensional_analogy-sublist" class="vector-toc-list"> <li id="toc-Cross-sections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cross-sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Cross-sections</span> </div> </a> <ul id="toc-Cross-sections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projections"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Projections</span> </div> </a> <ul id="toc-Projections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shadows" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shadows"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Shadows</span> </div> </a> <ul id="toc-Shadows-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bounding_regions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bounding_regions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Bounding regions</span> </div> </a> <ul id="toc-Bounding_regions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hypervolume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hypervolume"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Hypervolume</span> </div> </a> <ul id="toc-Hypervolume-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In culture</span> </div> </a> <button aria-controls="toc-In_culture-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In culture subsection</span> </button> <ul id="toc-In_culture-sublist" class="vector-toc-list"> <li id="toc-In_art" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_art"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>In art</span> </div> </a> <ul id="toc-In_art-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_literature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_literature"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>In literature</span> </div> </a> <ul id="toc-In_literature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_philosophy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_philosophy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>In philosophy</span> </div> </a> <ul id="toc-In_philosophy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Four-dimensional space</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 41 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-41" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">41 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D8%B1%D8%A8%D8%A7%D8%B9%D9%8A_%D8%A7%D9%84%D8%A3%D8%A8%D8%B9%D8%A7%D8%AF" title="فضاء رباعي الأبعاد – Arabic" lang="ar" hreflang="ar" data-title="فضاء رباعي الأبعاد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9A%E0%A6%A4%E0%A7%81%E0%A6%B0%E0%A7%8D%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="চতুর্মাত্রিক ক্ষেত্র – Bangla" lang="bn" hreflang="bn" data-title="চতুর্মাত্রিক ক্ষেত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D0%B2%D1%8A%D1%80%D1%82%D0%BE_%D0%B8%D0%B7%D0%BC%D0%B5%D1%80%D0%B5%D0%BD%D0%B8%D0%B5" title="Четвърто измерение – Bulgarian" lang="bg" hreflang="bg" data-title="Четвърто измерение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quarta_dimensi%C3%B3" title="Quarta dimensió – Catalan" lang="ca" hreflang="ca" data-title="Quarta dimensió" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%C4%83%D0%B2%D0%B0%D1%82%C4%83_%D1%85%D0%B0%D0%BF%D0%B0%D0%BB%D0%BB%C4%83_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Тăватă хапаллă уçлăх – Chuvash" lang="cv" hreflang="cv" data-title="Тăватă хапаллă уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C4%8Ctvrt%C3%BD_rozm%C4%9Br" title="Čtvrtý rozměr – Czech" lang="cs" hreflang="cs" data-title="Čtvrtý rozměr" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_pedwar_dimensiwn" title="Gofod pedwar dimensiwn – Welsh" lang="cy" hreflang="cy" data-title="Gofod pedwar dimensiwn" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/4D" title="4D – German" lang="de" hreflang="de" data-title="4D" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuarta_dimensi%C3%B3n" title="Cuarta dimensión – Spanish" lang="es" hreflang="es" data-title="Cuarta dimensión" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvara_dimensio" title="Kvara dimensio – Esperanto" lang="eo" hreflang="eo" data-title="Kvara dimensio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Laugarren_dimentsio" title="Laugarren dimentsio – Basque" lang="eu" hreflang="eu" data-title="Laugarren dimentsio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%DA%86%D9%87%D8%A7%D8%B1%D8%A8%D8%B9%D8%AF%DB%8C" title="فضای چهاربعدی – Persian" lang="fa" hreflang="fa" data-title="فضای چهاربعدی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://fr.wikipedia.org/wiki/Espace_%C3%A0_quatre_dimensions" title="Espace à quatre dimensions – French" lang="fr" hreflang="fr" data-title="Espace à quatre dimensions" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/4%EC%B0%A8%EC%9B%90" title="4차원 – Korean" lang="ko" hreflang="ko" data-title="4차원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%A1%D5%BC%D5%A1%D5%B9%D5%A1%D6%83_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Քառաչափ տարածություն – Armenian" lang="hy" hreflang="hy" data-title="Քառաչափ տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9A%E0%A4%A4%E0%A5%81%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%AE_%E0%A4%B8%E0%A4%AE%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%BF" title="चतुर्विम समष्टि – Hindi" lang="hi" hreflang="hi" data-title="चतुर्विम समष्टि" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_dimensi_empat" title="Ruang dimensi empat – Indonesian" lang="id" hreflang="id" data-title="Ruang dimensi empat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quarta_dimensione" title="Quarta dimensione – Italian" lang="it" hreflang="it" data-title="Quarta dimensione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%90%D7%A8%D7%91%D7%A2-%D7%9E%D7%9E%D7%93%D7%99" title="מרחב ארבע-ממדי – Hebrew" lang="he" hreflang="he" data-title="מרחב ארבע-ממדי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D3%A9%D1%80%D1%82_%D3%A9%D0%BB%D1%88%D0%B5%D0%BC%D0%B4%D1%96_%D0%BA%D0%B5%D2%A3%D1%96%D1%81%D1%82%D1%96%D0%BA" title="Төрт өлшемді кеңістік – Kazakh" lang="kk" hreflang="kk" data-title="Төрт өлшемді кеңістік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Negyedik_dimenzi%C3%B3" title="Negyedik dimenzió – Hungarian" lang="hu" hreflang="hu" data-title="Negyedik dimenzió" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D0%B8%D1%80%D0%B8%D0%B4%D0%B8%D0%BC%D0%B5%D0%BD%D0%B7%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Четиридимензионален простор – Macedonian" lang="mk" hreflang="mk" data-title="Четиридимензионален простор" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_empat_dimensi" title="Ruang empat dimensi – Malay" lang="ms" hreflang="ms" data-title="Ruang empat dimensi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vierde_dimensie" title="Vierde dimensie – Dutch" lang="nl" hreflang="nl" data-title="Vierde dimensie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/4%E6%AC%A1%E5%85%83" title="4次元 – Japanese" lang="ja" hreflang="ja" data-title="4次元" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Firedimensjonal" title="Firedimensjonal – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Firedimensjonal" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Firedimensjonal" title="Firedimensjonal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Firedimensjonal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Quarta_dimens%C3%A3o" title="Quarta dimensão – Portuguese" lang="pt" hreflang="pt" data-title="Quarta dimensão" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_cvadridimensional" title="Spațiu cvadridimensional – Romanian" lang="ro" hreflang="ro" data-title="Spațiu cvadridimensional" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B5%D1%82%D1%8B%D1%80%D1%91%D1%85%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Четырёхмерное пространство – Russian" lang="ru" hreflang="ru" data-title="Четырёхмерное пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_kat%C3%ABr-dimensionale" title="Hapësira katër-dimensionale – Albanian" lang="sq" hreflang="sq" data-title="Hapësira katër-dimensionale" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/4D" title="4D – Simple English" lang="en-simple" hreflang="en-simple" data-title="4D" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a 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#a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Four-dimensional_space" title="Special:EditPage/Four-dimensional space">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Four-dimensional+space%22">"Four-dimensional space"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Four-dimensional+space%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Four-dimensional+space%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Four-dimensional+space%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Four-dimensional+space%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Four-dimensional+space%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Frame"><a href="/wiki/File:8-cell-simple.gif" class="mw-file-description"><img alt="Animation of a transforming tesseract or 4-cube" src="//upload.wikimedia.org/wikipedia/commons/5/55/8-cell-simple.gif" decoding="async" width="256" height="256" class="mw-file-element" data-file-width="256" data-file-height="256" /></a><figcaption>The 4D equivalent of a <a href="/wiki/Cube" title="Cube">cube</a> is known as a <a href="/wiki/Tesseract" title="Tesseract">tesseract</a>, seen rotating here in four-dimensional space, yet 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nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size notpageimage" typeof="mw:File/Frameless"><a href="/wiki/File:Stereographic_projection_in_3D.svg" class="mw-file-description"><img alt="Stereographic projection from the top of a sphere onto a plane beneath it" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/220px-Stereographic_projection_in_3D.svg.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/330px-Stereographic_projection_in_3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/440px-Stereographic_projection_in_3D.svg.png 2x" data-file-width="870" data-file-height="639" /></a></span><div class="sidebar-caption"><a href="/wiki/Projective_geometry" title="Projective geometry">Projecting</a> a <a href="/wiki/Sphere" title="Sphere">sphere</a> to a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a> <ul><li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li> <li><a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a> <ul><li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a> <ul><li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a> <ul><li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital</a></li></ul></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence </a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a> <ul><li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="/wiki/Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a> <ul><li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a></li> <li><a href="/wiki/Curve" title="Curve">Curve</a></li> <li><a href="/wiki/Diagonal" title="Diagonal">Diagonal</a></li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a>)</li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Line_(geometry)" title="Line (geometry)">Line</a> <ul><li><a href="/wiki/Line_segment" title="Line segment">segment</a></li> <li><a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li> <li><a href="/wiki/Length" title="Length">Length</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li> <li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li> <li><a href="/wiki/Hypotenuse" title="Hypotenuse">Hypotenuse</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Diameter" title="Diameter">Diameter</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Volume" title="Volume">Volume</a></li></ul> <ul><li><a href="/wiki/Cube" title="Cube">Cube</a> <ul><li><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></li></ul></li> <li><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li> <li><a href="/wiki/Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a class="mw-selflink selflink">Four</a>- / other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li> <li><a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;"> <a href="/wiki/List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1–1400s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1400s–1700s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1700s–1900s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Present day</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_geometry" title="Template:General geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_geometry" title="Template talk:General geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_geometry" title="Special:EditPage/Template:General geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Four-dimensional space</b> (<b>4D</b>) is the mathematical extension of the concept of <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called <i><a href="/wiki/Dimension" title="Dimension">dimensions</a></i>, to describe the <a href="/wiki/Size" title="Size">sizes</a> or <a href="/wiki/Location" title="Location">locations</a> of objects in the everyday world. For example, the <a href="/wiki/Volume" title="Volume">volume</a> of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="texhtml mvar" style="font-style:italic;">z</span>). This concept of ordinary space is called <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> because it corresponds to <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclid's geometry</a>, which was originally abstracted from the spatial experiences of everyday life. </p><p>The idea of adding a fourth dimension appears in <a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a>'s "Dimensions", published in 1754,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> but the mathematics of more than three dimensions only <a href="/wiki/Euclidean_geometry#Higher_dimensions" title="Euclidean geometry">emerged in the 19th century</a>. The general concept of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> with any number of dimensions was fully developed by the Swiss mathematician <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> before 1853. Schläfli's work received little attention during his lifetime and was published only posthumously, in 1901,<sup id="cite_ref-FOOTNOTESchläfli1901_2-0" class="reference"><a href="#cite_note-FOOTNOTESchläfli1901-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> but meanwhile the fourth Euclidean dimension was rediscovered by others. In 1880 <a href="/wiki/Charles_Howard_Hinton" title="Charles Howard Hinton">Charles Howard Hinton</a> popularized it in an essay, "<a href="https://en.wikisource.org/wiki/What_is_the_Fourth_Dimension%3F" class="extiw" title="s:What is the Fourth Dimension?">What is the Fourth Dimension?</a>", in which he explained the concept of a "<a href="/wiki/Four-dimensional_cube" class="mw-redirect" title="Four-dimensional cube">four-dimensional cube</a>" with a step-by-step generalization of the properties of lines, squares, and cubes. The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The eight lines connecting the vertices of the two cubes in this case represent a <i>single direction</i> in the "unseen" fourth dimension. </p><p>Higher-dimensional spaces (greater than three) have since become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without using such spaces. <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>'s <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> is formulated in 4D space,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> although not in a Euclidean 4D space. Einstein's concept of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> has a <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski structure</a> based on a <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of <a href="/wiki/Regular_4-polytope" title="Regular 4-polytope">Schläfli's Euclidean 4D space</a>. </p><p>Single locations in Euclidean 4D space can be given as <a href="/wiki/Vector_space" title="Vector space">vectors</a> or <i><a href="/wiki/N-tuples" class="mw-redirect" title="N-tuples">4-tuples</a></i>, i.e., as ordered lists of numbers such as <span class="texhtml">(<i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>)</span>. It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of higher-dimensional spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible <a href="/wiki/Regular_4-polytope" title="Regular 4-polytope">regular 4D objects</a>, the <a href="/wiki/Tesseract" title="Tesseract">tesseract</a>, which is <a href="/wiki/Hypercube" title="Hypercube">analogous</a> to the 3D <a href="/wiki/Cube" title="Cube">cube</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a> wrote in his <span title="French-language text"><i lang="fr">Mécanique analytique</i></span> (published 1788, based on work done around 1755) that <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> can be viewed as operating in a four-dimensional space— three dimensions of space, and one of time.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> As early as 1827, <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a> realized that a fourth <i>spatial</i> dimension would allow a three-dimensional form to be rotated onto its mirror-image.<sup id="cite_ref-FOOTNOTECoxeter1973141§7.x._Historical_remarks_5-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973141§7.x._Historical_remarks-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The general concept of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> with any number of dimensions was fully developed by the Swiss mathematician <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> in the mid-19th century, at a time when <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a>, <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Grassman</a> and <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a> were the only other people who had ever conceived the possibility of geometry in more than three dimensions.<sup id="cite_ref-FOOTNOTECoxeter1973141–144§7._Ordinary_Polytopes_in_Higher_Space;_§7.x._Historical_remarks_6-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973141–144§7._Ordinary_Polytopes_in_Higher_Space;_§7.x._Historical_remarks-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> By 1853 Schläfli had discovered all the regular <a href="/wiki/Polytope" title="Polytope">polytopes</a> that exist in higher dimensions, including the <a href="/wiki/Regular_4-polytope" title="Regular 4-polytope">four-dimensional analogs</a> of the <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a>. </p><p>An arithmetic of four spatial dimensions, called <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>, was defined by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> in 1843. This <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> was the source of the science of <a href="/wiki/Vector_analysis" class="mw-redirect" title="Vector analysis">vector analysis</a> in three dimensions as recounted by <a href="/wiki/Michael_J._Crowe" title="Michael J. Crowe">Michael J. Crowe</a> in <i><a href="/wiki/A_History_of_Vector_Analysis" title="A History of Vector Analysis">A History of Vector Analysis</a></i>. Soon after, <a href="/wiki/Tessarine" class="mw-redirect" title="Tessarine">tessarines</a> and <a href="/wiki/Coquaternion" class="mw-redirect" title="Coquaternion">coquaternions</a> were introduced as other four-dimensional <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebras over <b>R</b></a>. In 1886, <a href="/wiki/Victor_Schlegel" title="Victor Schlegel">Victor Schlegel</a> described<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> his method of visualizing <a href="/wiki/4-polytope" title="4-polytope">four-dimensional objects</a> with <a href="/wiki/Schlegel_diagram" title="Schlegel diagram">Schlegel diagrams</a>. </p><p>One of the first popular expositors of the fourth dimension was <a href="/wiki/Charles_Howard_Hinton" title="Charles Howard Hinton">Charles Howard Hinton</a>, starting in 1880 with his essay <i>What is the Fourth Dimension?</i>, published in the <a href="/wiki/Dublin_University" class="mw-redirect" title="Dublin University">Dublin University</a> magazine.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> He coined the terms <i><a href="/wiki/Tesseract" title="Tesseract">tesseract</a></i>, <i>ana</i> and <i>kata</i> in his book <i><a href="/wiki/A_New_Era_of_Thought" title="A New Era of Thought">A New Era of Thought</a></i> and introduced a method for visualizing the fourth dimension using cubes in the book <i>Fourth Dimension</i>.<sup id="cite_ref-Hinton_9-0" class="reference"><a href="#cite_note-Hinton-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> in his January 1962 "<a href="/wiki/Mathematical_Games_column" class="mw-redirect" title="Mathematical Games column">Mathematical Games column</a>" in <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. </p><p>Higher dimensional non-Euclidean spaces were put on a firm footing by <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>'s 1854 <a href="/wiki/Habilitationsschrift" class="mw-redirect" title="Habilitationsschrift">thesis</a>, <span title="German-language text"><i lang="de">Über die Hypothesen welche der Geometrie zu Grunde liegen</i></span>, in which he considered a "point" to be any sequence of coordinates <span class="texhtml">(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>)</span>. In 1908, <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> presented a paper<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> consolidating the role of time as the fourth dimension of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>, the basis for <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein's</a> theories of <a href="/wiki/Special_relativity" title="Special relativity">special</a> and <a href="/wiki/General_relativity" title="General relativity">general relativity</a>.<sup id="cite_ref-Møller_12-0" class="reference"><a href="#cite_note-Møller-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> But the geometry of spacetime, being <a href="/wiki/Non-Euclidean" class="mw-redirect" title="Non-Euclidean">non-Euclidean</a>, is profoundly different from that explored by Schläfli and popularised by Hinton. The study of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> required Riemann's mathematics which is quite different from that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 <a href="/wiki/H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">H. S. M. Coxeter</a> felt compelled to write: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Little, if anything, is gained by representing the fourth Euclidean dimension as <i>time</i>. In fact, this idea, so attractively developed by <a href="/wiki/The_Time_Machine" title="The Time Machine">H. G. Wells in <i>The Time Machine</i></a>, has led such authors as <a href="/wiki/An_Experiment_with_Time" title="An Experiment with Time">John William Dunne (<i>An Experiment with Time</i>)</a> into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is <i>not</i> Euclidean, and consequently has no connection with the present investigation. </p><div class="templatequotecite">— <cite><a href="/wiki/H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">H. S. M. Coxeter</a>, <i>Regular Polytopes</i><sup id="cite_ref-FOOTNOTECoxeter1973119_13-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973119-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <div class="mw-heading mw-heading2"><h2 id="Vectors">Vectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=2" title="Edit section: Vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematically, a four-dimensional space is a <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a> that needs four parameters to specify a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> in it. For example, a general point might have position <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> <span class="texhtml"><b>a</b></span>, equal to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811f32554d9d0fa0aad7ea7d8c71b2e6b14dc540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:12.147ex; height:12.509ex;" alt="{\displaystyle \mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}.}"></span></dd></dl> <p>This can be written in terms of the four <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> vectors <span class="texhtml">(<b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, <b>e</b><sub>3</sub>, <b>e</b><sub>4</sub>)</span>, given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\mathbf {e} _{2}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\mathbf {e} _{3}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\mathbf {e} _{4}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{1}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\mathbf {e} _{2}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\mathbf {e} _{3}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\mathbf {e} _{4}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5f1b032f59777961ff9c0d04952c8f9f02ba50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:49.183ex; height:12.509ex;" alt="{\displaystyle \mathbf {e} _{1}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\mathbf {e} _{2}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\mathbf {e} _{3}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\mathbf {e} _{4}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},}"></span></dd></dl> <p>so the general vector <span class="texhtml"><b>a</b></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}+a_{4}\mathbf {e} _{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}+a_{4}\mathbf {e} _{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de55c426e0abd763837efab6ae517b3dcd3e3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.82ex; height:2.343ex;" alt="{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}+a_{4}\mathbf {e} _{4}.}"></span></dd></dl> <p>Vectors add, subtract and scale as in three dimensions. </p><p>The <a href="/wiki/Dot_product" title="Dot product">dot product</a> of Euclidean three-dimensional space generalizes to four dimensions as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400fa06d6a7e26fdd7215c2e7fc368c8b0f0e618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.074ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.}"></span></dd></dl> <p>It can be used to calculate the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> or <a href="/wiki/Euclidean_distance" title="Euclidean distance">length</a> of a vector, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\mathbf {a} \right|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mathbf {a} \right|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ca31fa2e02b5444470c474b49e9684cfdc4933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.632ex; height:4.843ex;" alt="{\displaystyle \left|\mathbf {a} \right|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}},}"></span></dd></dl> <p>and calculate or define the <a href="/wiki/Angle" title="Angle">angle</a> between two non-zero vectors as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{\left|\mathbf {a} \right|\left|\mathbf {b} \right|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{\left|\mathbf {a} \right|\left|\mathbf {b} \right|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce76522d12183df70f2af70626b1b03ce1812a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.036ex; height:6.176ex;" alt="{\displaystyle \theta =\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{\left|\mathbf {a} \right|\left|\mathbf {b} \right|}}.}"></span></dd></dl> <p>Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate <a href="/wiki/Pairing" title="Pairing">pairing</a> different from the dot product: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}-a_{4}b_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}-a_{4}b_{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086ffc6f1f2cf329c62639dbeedd0f816f3ae55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.074ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}-a_{4}b_{4}.}"></span></dd></dl> <p>As an example, the distance squared between the points <span class="texhtml">(0,0,0,0)</span> and <span class="texhtml">(1,1,1,0)</span> is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between <span class="texhtml">(0,0,0,0)</span> and <span class="texhtml">(1,1,1,1)</span> is 4 in Euclidean space and 2 in Minkowski space; increasing <span class="texhtml"><i>b</i><sub>4</sub></span> decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity. </p><p>The <a href="/wiki/Cross_product" title="Cross product">cross product</a> is not defined in four dimensions. Instead, the <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a> is used for some applications, and is defined as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {a} \wedge \mathbf {b} =(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{12}+(a_{1}b_{3}-a_{3}b_{1})\mathbf {e} _{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf {e} _{14}+(a_{2}b_{3}-a_{3}b_{2})\mathbf {e} _{23}\\+(a_{2}b_{4}-a_{4}b_{2})\mathbf {e} _{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {a} \wedge \mathbf {b} =(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{12}+(a_{1}b_{3}-a_{3}b_{1})\mathbf {e} _{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf {e} _{14}+(a_{2}b_{3}-a_{3}b_{2})\mathbf {e} _{23}\\+(a_{2}b_{4}-a_{4}b_{2})\mathbf {e} _{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0242d12a3a4bde0af48b24a09480b0ef2ebbce89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:83.43ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {a} \wedge \mathbf {b} =(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{12}+(a_{1}b_{3}-a_{3}b_{1})\mathbf {e} _{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf {e} _{14}+(a_{2}b_{3}-a_{3}b_{2})\mathbf {e} _{23}\\+(a_{2}b_{4}-a_{4}b_{2})\mathbf {e} _{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.\end{aligned}}}"></span></dd></dl> <p>This is <a href="/wiki/Bivector" title="Bivector">bivector</a> valued, with bivectors in four dimensions forming a <a href="/wiki/Six-dimensional_space" title="Six-dimensional space">six-dimensional</a> linear space with basis <span class="texhtml">(<b>e</b><sub>12</sub>, <b>e</b><sub>13</sub>, <b>e</b><sub>14</sub>, <b>e</b><sub>23</sub>, <b>e</b><sub>24</sub>, <b>e</b><sub>34</sub>)</span>. They can be used to generate rotations in four dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Orthogonality_and_vocabulary">Orthogonality and vocabulary</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=3" title="Edit section: Orthogonality and vocabulary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the familiar three-dimensional space of daily life, there are three <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate axes</a>—usually labeled <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="texhtml mvar" style="font-style:italic;">z</span>—with each axis <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called <i>up</i>, <i>down</i>, <i>east</i>, <i>west</i>, <i>north</i>, and <i>south</i>. Positions along these axes can be called <i>altitude</i>, <i>longitude</i>, and <i>latitude</i>. Lengths measured along these axes can be called <i>height</i>, <i>width</i>, and <i>depth</i>. </p><p>Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled <span class="texhtml mvar" style="font-style:italic;">w</span>. To describe the two additional cardinal directions, <a href="/wiki/Charles_Howard_Hinton" title="Charles Howard Hinton">Charles Howard Hinton</a> coined the terms <i>ana</i> and <i>kata</i>, from the Greek words meaning "up toward" and "down from", respectively.<sup id="cite_ref-Hinton_9-1" class="reference"><a href="#cite_note-Hinton-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 160">: 160 </span></sup> </p><p>As mentioned above, Hermann Minkowski exploited the idea of four dimensions to discuss cosmology including the finite <a href="/wiki/Velocity_of_light" class="mw-redirect" title="Velocity of light">velocity of light</a>. In appending a time dimension to three-dimensional space, he specified an alternative perpendicularity, <a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">hyperbolic orthogonality</a>. This notion provides his four-dimensional space with a modified <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">simultaneity</a> appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional <a href="/wiki/Absolute_space_and_time" title="Absolute space and time">absolute space and time</a> cosmology previously used in a universe of three space dimensions and one time dimension. </p> <div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=4" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space" title="Rotations in 4-dimensional Euclidean space">Rotations in 4-dimensional Euclidean space</a></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Four-dimensional_space" title="Special:EditPage/Four-dimensional space">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Four-dimensional+space%22">"Four-dimensional space"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Four-dimensional+space%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Four-dimensional+space%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Four-dimensional+space%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Four-dimensional+space%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Four-dimensional+space%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">November 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom. </p><p>Just as in three dimensions there are <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> made of two dimensional <a href="/wiki/Polygon" title="Polygon">polygons</a>, in four dimensions there are polychora made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>. In four dimensions, there are 6 <a href="/wiki/Convex_regular_4-polytope" class="mw-redirect" title="Convex regular 4-polytope">convex regular 4-polytopes</a>, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex <a href="/wiki/Uniform_4-polytope" title="Uniform 4-polytope">uniform 4-polytopes</a>, analogous to the 13 semi-regular <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a> in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes. </p> <table class="wikitable"> <caption>Regular polytopes in four dimensions<br />(Displayed as orthogonal projections in each <a href="/wiki/Coxeter_plane" class="mw-redirect" title="Coxeter plane">Coxeter plane</a> of symmetry) </caption> <tbody><tr> <th>A<sub>4</sub>, [3,3,3] </th> <th colspan="2">B<sub>4</sub>, [4,3,3] </th> <th>F<sub>4</sub>, [3,4,3] </th> <th colspan="2">H<sub>4</sub>, [5,3,3] </th></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:4-simplex_t0.svg" class="mw-file-description" title="altN=4-simplex"><img alt="altN=4-simplex" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/120px-4-simplex_t0.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/180px-4-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/240px-4-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span><br /><a href="/wiki/5-cell" title="5-cell">5-cell</a><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{3,3,3} </td> <td><span typeof="mw:File"><a href="/wiki/File:4-cube_t0.svg" class="mw-file-description" title="altN=4-cube"><img alt="altN=4-cube" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/120px-4-cube_t0.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/180px-4-cube_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/240px-4-cube_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span><br /><a href="/wiki/Tesseract" title="Tesseract">tesseract</a><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{4,3,3} </td> <td><span typeof="mw:File"><a href="/wiki/File:4-cube_t3.svg" class="mw-file-description" title="altN=4-orthoplex"><img alt="altN=4-orthoplex" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/120px-4-cube_t3.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/180px-4-cube_t3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/240px-4-cube_t3.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span><br /><a href="/wiki/16-cell" title="16-cell">16-cell</a><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{3,3,4} </td> <td><span typeof="mw:File"><a href="/wiki/File:24-cell_graph.svg" class="mw-file-description" title="altN=24-cell"><img alt="altN=24-cell" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/24-cell_graph.svg/120px-24-cell_graph.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/24-cell_graph.svg/180px-24-cell_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/24-cell_graph.svg/240px-24-cell_graph.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/24-cell" title="24-cell">24-cell</a><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{3,4,3} </td> <td><span typeof="mw:File"><a href="/wiki/File:600-cell_graph_H4.svg" class="mw-file-description" title="altN=600-cell"><img alt="altN=600-cell" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/120px-600-cell_graph_H4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/180px-600-cell_graph_H4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/240px-600-cell_graph_H4.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/600-cell" title="600-cell">600-cell</a><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{3,3,5} </td> <td><span typeof="mw:File"><a href="/wiki/File:120-cell_graph_H4.svg" class="mw-file-description" title="altN=120-cell"><img alt="altN=120-cell" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/120px-120-cell_graph_H4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/180px-120-cell_graph_H4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/240px-120-cell_graph_H4.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/120-cell" title="120-cell">120-cell</a><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{5,3,3} </td></tr></tbody></table> <p>In three dimensions, a circle may be <a href="/wiki/Extrude" class="mw-redirect" title="Extrude">extruded</a> to form a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a>. In four dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps", known as a <a href="/wiki/Spherinder" title="Spherinder">spherinder</a>), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2022)">citation needed</span></a></i>]</sup> The <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of two circles may be taken to obtain a <a href="/wiki/Duocylinder" title="Duocylinder">duocylinder</a>. All three can "roll" in four-dimensional space, each with its properties. </p><p>In three dimensions, curves can form <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a> but surfaces cannot (unless they are self-intersecting). In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction—but 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (December 2016)">page needed</span></a></i>]</sup> Because these surfaces are two-dimensional, they can form much more complex knots than strings in 3D space can. The <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> is an example of such a knotted surface.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2013)">citation needed</span></a></i>]</sup> Another such surface is the <a href="/wiki/Real_projective_plane" title="Real projective plane">real projective plane</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2013)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hypersphere">Hypersphere</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=5" title="Edit section: Hypersphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Clifford-torus.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Clifford-torus.gif" decoding="async" width="255" height="255" class="mw-file-element" data-file-width="255" data-file-height="255" /></a><figcaption><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> of a <a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a>: the set of points <span class="texhtml">(cos(<i>a</i>), sin(<i>a</i>), cos(<i>b</i>), sin(<i>b</i>))</span>, which is a subset of the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></div> <p>The set of points in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean 4-space</a> having the same distance <span class="texhtml mvar" style="font-style:italic;">R</span> from a fixed point <span class="texhtml"><i>P</i><sub>0</sub></span> forms a <a href="/wiki/Hypersurface" title="Hypersurface">hypersurface</a> known as a <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a>. The hyper-volume of the enclosed space is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V} ={\begin{matrix}{\frac {1}{2}}\end{matrix}}\pi ^{2}R^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} ={\begin{matrix}{\frac {1}{2}}\end{matrix}}\pi ^{2}R^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9515b3e41b26326ca4f7fe7c7f9fe7baef326662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.968ex; margin-bottom: -0.203ex; width:12.734ex; height:3.509ex;" alt="{\displaystyle \mathbf {V} ={\begin{matrix}{\frac {1}{2}}\end{matrix}}\pi ^{2}R^{4}}"></span></dd></dl> <p>This is part of the <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker metric</a> in <a href="/wiki/General_relativity" title="General relativity">General relativity</a> where <span class="texhtml mvar" style="font-style:italic;">R</span> is substituted by function <span class="texhtml"><i>R</i>(<i>t</i>)</span> with <span class="texhtml mvar" style="font-style:italic;">t</span> meaning the cosmological age of the universe. Growing or shrinking <span class="texhtml mvar" style="font-style:italic;">R</span> with time means expanding or collapsing universe, depending on the mass density inside.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Four-dimensional_perception_in_humans">Four-dimensional perception in humans</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=6" title="Edit section: Four-dimensional perception in humans"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Research using <a href="/wiki/Virtual_reality" title="Virtual reality">virtual reality</a> finds that humans, despite living in a three-dimensional world, can, without special practice, make spatial judgments about line segments embedded in four-dimensional space, based on their length (one-dimensional) and the angle (two-dimensional) between them.<sup id="cite_ref-Ambinder_16-0" class="reference"><a href="#cite_note-Ambinder-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments".<sup id="cite_ref-Ambinder_16-1" class="reference"><a href="#cite_note-Ambinder-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In another study,<sup id="cite_ref-Aflalo_17-0" class="reference"><a href="#cite_note-Aflalo-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> the ability of humans to orient themselves in 2D, 3D, and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops (i.e. actually <a href="/wiki/Labyrinth" title="Labyrinth">labyrinths</a>). The graphical interface was based on John McIntosh's free 4D Maze game.<sup id="cite_ref-McIntosh_18-0" class="reference"><a href="#cite_note-McIntosh-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D (the lower-dimensional cases were for comparison and for the participants to learn the method). </p><p>However, a 2020 review underlined how these studies are composed of a small subject sample and mainly of college students. It also pointed out other issues that future research has to resolve: elimination of <a href="/wiki/Artifact_(error)" title="Artifact (error)">artifacts</a> (these could be caused, for example, by strategies to resolve the required task that don't use 4D representation/4D reasoning and feedback given by researchers to speed up the adaptation process) and analysis on inter-subject variability (if 4D perception is possible, its acquisition could be limited to a subset of humans, to a specific <a href="/wiki/Critical_period" title="Critical period">critical period</a>, or to people's attention or motivation). Furthermore, it is undetermined if there is a more appropriate way to project the 4-dimension (because there are no restrictions on how the 4-dimension can be projected). Researchers also hypothesized that human acquisition of 4D perception could result in the activation of brain visual areas and <a href="/wiki/Entorhinal_cortex" title="Entorhinal cortex">entorhinal cortex</a>. If so they suggest that it could be used as a strong indicator of 4D space perception acquisition. Authors also suggested using a variety of different <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural network architectures</a> (with different <i><a href="/wiki/A_priori" class="mw-redirect" title="A priori">a priori</a></i> assumptions) to understand which ones are or are not able to learn.<sup id="cite_ref-Ogmen_19-0" class="reference"><a href="#cite_note-Ogmen-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dimensional_analogy">Dimensional analogy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=7" title="Edit section: Dimensional analogy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tesseract_net.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Tesseract_net.svg/220px-Tesseract_net.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Tesseract_net.svg/330px-Tesseract_net.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Tesseract_net.svg/440px-Tesseract_net.svg.png 2x" data-file-width="451" data-file-height="341" /></a><figcaption>A net of a tesseract</figcaption></figure> <p>To understand the nature of four-dimensional space, a device called <i>dimensional analogy</i> is commonly employed. Dimensional analogy is the study of how (<span class="texhtml"><i>n</i> − 1</span>) dimensions relate to <span class="texhtml mvar" style="font-style:italic;">n</span> dimensions, and then inferring how <span class="texhtml mvar" style="font-style:italic;">n</span> dimensions would relate to (<span class="texhtml"><i>n</i> + 1</span>) dimensions.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The dimensional analogy was used by <a href="/wiki/Edwin_Abbott_Abbott" title="Edwin Abbott Abbott">Edwin Abbott Abbott</a> in the book <i><a href="/wiki/Flatland" title="Flatland">Flatland</a></i>, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like powers, such as ability to remove objects from a safe without breaking it open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension. </p><p>By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from the three-dimensional perspective. <a href="/wiki/Rudy_Rucker" title="Rudy Rucker">Rudy Rucker</a> illustrates this in his novel <i><a href="/wiki/Spaceland_(novel)" title="Spaceland (novel)">Spaceland</a></i>, in which the protagonist encounters four-dimensional beings who demonstrate such powers. </p> <div class="mw-heading mw-heading3"><h3 id="Cross-sections">Cross-sections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=8" title="Edit section: Cross-sections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-section</a> of the three-dimensional object within this plane. For example, if a sphere passed through a sheet of paper, beings in the paper would see first a single point. A circle gradually grows larger, until it reaches the diameter of the sphere, and then gets smaller again, until it shrinks to a point and disappears. The 2D beings would not see a circle in the same way as three-dimensional beings do; rather, they only see a <a href="/wiki/One-dimensional_space" title="One-dimensional space">one-dimensional</a> projection of the circle on their 1D "retina". Similarly, if a four-dimensional object passed through a three-dimensional (hyper) surface, one could observe a three-dimensional cross-section of the four-dimensional object. For example, a <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a> would appear first as a point, then as a growing sphere (until it reaches the "hyperdiameter" of the hypersphere), with the sphere then shrinking to a single point and then disappearing.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This means of visualizing aspects of the fourth dimension was used in the novel <i>Flatland</i> and also in several works of <a href="/wiki/Charles_Howard_Hinton" title="Charles Howard Hinton">Charles Howard Hinton</a>.<sup id="cite_ref-Hinton_9-2" class="reference"><a href="#cite_note-Hinton-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 11–14">: 11–14 </span></sup> And, in the same way, three-dimensional beings (such as humans with a 2D retina) can see all the sides and the insides of a 2D shape simultaneously, a 4D being could see all faces and the inside of a 3D shape at once with their 3D retina. </p> <div class="mw-heading mw-heading3"><h3 id="Projections">Projections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=9" title="Edit section: Projections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A useful application of dimensional analogy in visualizing higher dimensions is in <a href="/wiki/Graphical_projection" class="mw-redirect" title="Graphical projection">projection</a>. A projection is a way of representing an <i>n</i>-dimensional object in <span class="texhtml"><i>n</i> − 1</span> dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places, and things are represented in two dimensions by projecting the objects onto a flat surface. By doing this, the dimension orthogonal to the screen (<i>depth</i>) is removed and replaced with indirect information. The <a href="/wiki/Retina" title="Retina">retina</a> of the <a href="/wiki/Human_eye" title="Human eye">eye</a> is also a two-dimensional <a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">array</a> of <a href="/wiki/Sensory_receptor" class="mw-redirect" title="Sensory receptor">receptors</a> but the <a href="/wiki/Brain" title="Brain">brain</a> can perceive the nature of three-dimensional objects by inference from indirect information (such as shading, <a href="/wiki/Foreshortening" class="mw-redirect" title="Foreshortening">foreshortening</a>, <a href="/wiki/Binocular_vision" title="Binocular vision">binocular vision</a>, etc.). <a href="/wiki/Artist" title="Artist">Artists</a> often use <a href="/wiki/Perspective_(graphical)" title="Perspective (graphical)">perspective</a> to give an illusion of three-dimensional depth to two-dimensional pictures. The <i>shadow</i>, cast by a fictitious grid model of a rotating tesseract on a plane surface, as shown in the figures, is also the result of projections. </p><p>Similarly, objects in the fourth dimension can be mathematically projected to the familiar three dimensions, where they can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina. </p><p>The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying dimensional analogy, one may infer four-dimensional "depth" from these effects. </p><p>As an illustration of this principle, the following sequence of images compares various views of the three-dimensional <a href="/wiki/Cube" title="Cube">cube</a> with analogous projections of the four-dimensional tesseract into three-dimensional space. </p> <table class="wiki table"> <tbody><tr> <th>Cube </th> <th>Tesseract </th> <th>Description </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Cube-face-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Cube-face-first.png/160px-Cube-face-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Cube-face-first.png/240px-Cube-face-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9a/Cube-face-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tesseract-perspective-cell-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Tesseract-perspective-cell-first.png/160px-Tesseract-perspective-cell-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Tesseract-perspective-cell-first.png/240px-Tesseract-perspective-cell-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/fd/Tesseract-perspective-cell-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td>The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the <b>cell-first perspective projection</b>, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube. <p>Note that the other 5 faces of the cube are not seen here. They are <i>obscured</i> by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell. </p> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Cube-edge-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Cube-edge-first.png/160px-Cube-edge-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Cube-edge-first.png/240px-Cube-edge-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/38/Cube-edge-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tesseract-perspective-face-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Tesseract-perspective-face-first.png/160px-Tesseract-perspective-face-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Tesseract-perspective-face-first.png/240px-Tesseract-perspective-face-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7e/Tesseract-perspective-face-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td>The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the <b>face-first perspective projection</b>, shown on the right. Just as the edge-first projection of the cube consists of two <a href="/wiki/Trapezoid" title="Trapezoid">trapezoids</a>, the face-first projection of the tesseract consists of two <a href="/wiki/Frustum" title="Frustum">frustums</a>. <p>The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells. </p> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Cube-vertex-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Cube-vertex-first.png/160px-Cube-vertex-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Cube-vertex-first.png/240px-Cube-vertex-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/51/Cube-vertex-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tesseract-perspective-edge-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Tesseract-perspective-edge-first.png/160px-Tesseract-perspective-edge-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Tesseract-perspective-edge-first.png/240px-Tesseract-perspective-edge-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/98/Tesseract-perspective-edge-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td>On the left is the cube viewed corner-first. This is analogous to the <b>edge-first perspective projection</b> of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 <a href="/wiki/Kite_(geometry)" title="Kite (geometry)">deltoids</a> surrounding a vertex, the tesseract's edge-first projection consists of 3 <a href="/wiki/Hexahedron" title="Hexahedron">hexahedral</a> volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet. </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Cube-edge-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Cube-edge-first.png/160px-Cube-edge-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Cube-edge-first.png/240px-Cube-edge-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/38/Cube-edge-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tesseract-perspective-edge-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Tesseract-perspective-edge-first.png/160px-Tesseract-perspective-edge-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Tesseract-perspective-edge-first.png/240px-Tesseract-perspective-edge-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/98/Tesseract-perspective-edge-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td>A different analogy may be drawn between the edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has <i>three</i> hexahedral volumes surrounding an edge. </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Cube-vertex-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Cube-vertex-first.png/160px-Cube-vertex-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Cube-vertex-first.png/240px-Cube-vertex-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/51/Cube-vertex-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tesseract-perspective-vertex-first.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Tesseract-perspective-vertex-first.png/160px-Tesseract-perspective-vertex-first.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Tesseract-perspective-vertex-first.png/240px-Tesseract-perspective-vertex-first.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a9/Tesseract-perspective-vertex-first.png 2x" data-file-width="320" data-file-height="240" /></a></span> </td> <td>On the left is the cube viewed corner-first. The <b>vertex-first perspective projection</b> of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first projection has <i>four</i> hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on the boundary of the projected volume, but at its center <i>inside</i>, where all four cells meet. <p>Only three of the cube's six faces can be seen here, because the other three faces lie <i>behind</i> these three faces, on the opposite side of the cube. Similarly, only four of the tesseract's eight cells can be seen here; the remaining four lie <i>behind</i> these four in the fourth direction, on the far side of the tesseract. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Shadows">Shadows</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=10" title="Edit section: Shadows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A concept closely related to projection is the casting of shadows. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_8-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/200px-Schlegel_wireframe_8-cell.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/300px-Schlegel_wireframe_8-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/400px-Schlegel_wireframe_8-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption></figcaption></figure> <p>If a light is shone on a three-dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shining on a four-dimensional object in a four-dimensional world would cast a three-dimensional shadow. </p><p>If the wireframe of a cube is lit from above, the resulting shadow on a flat two-dimensional surface is a square within a square with the corresponding corners connected. Similarly, if the wireframe of a tesseract were lit from "above" (in the fourth dimension), its shadow would be that of a three-dimensional cube within another three-dimensional cube suspended in midair (a "flat" surface from a four-dimensional perspective). (Note that, technically, the visual representation shown here is a two-dimensional image of the three-dimensional shadow of the four-dimensional wireframe figure.) </p> <div class="mw-heading mw-heading3"><h3 id="Bounding_regions">Bounding regions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=11" title="Edit section: Bounding regions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dimensional analogy also helps in inferring basic properties of objects in higher dimensions, such as the <a href="/wiki/Bounding_region" class="mw-redirect" title="Bounding region">bounding region</a>. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. </p><p>By applying dimensional analogy, one may infer that a four-dimensional cube, known as a <i><a href="/wiki/Tesseract" title="Tesseract">tesseract</a></i>, is bounded by three-dimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to <i>volumes</i> in the image, not merely two-dimensional surfaces. </p> <div class="mw-heading mw-heading3"><h3 id="Hypervolume">Hypervolume</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=12" title="Edit section: Hypervolume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>4-volume</b> or <a href="/wiki/Hypervolume" class="mw-redirect" title="Hypervolume">hypervolume</a> in 4D can be calculated in closed form for simple geometrical figures, such as the tesseract (<i>s</i><sup>4</sup>, for side length <i>s</i>) and the <a href="/wiki/4-ball_(mathematics)" class="mw-redirect" title="4-ball (mathematics)">4-ball</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{2}r^{4}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{2}r^{4}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a7d7fb6e81bd51c18f3d692241ed35468250fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.816ex; height:3.176ex;" alt="{\displaystyle \pi ^{2}r^{4}/2}"></span> for radius <i>r</i>). </p><p>Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the area enclosed by a circle in two dimensions (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f7b7f93f93e7ba7bebb97efbe88e181ce332e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.276ex; height:2.676ex;" alt="{\displaystyle A=\pi r^{2}}"></span>) and the volume enclosed by a sphere in three dimensions (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V={\frac {4}{3}}\pi r^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle V={\frac {4}{3}}\pi r^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65201b0d29e7addbc0ea7261f294dc432d5a4c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.979ex; height:3.676ex;" alt="{\textstyle V={\frac {4}{3}}\pi r^{3}}"></span>). One might guess that the volume enclosed by the sphere in four-dimensional space is a rational multiple of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi r^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fcfaaa994a8252f9bbc76436b3df39b5ec77c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.676ex;" alt="{\displaystyle \pi r^{4}}"></span>, but the correct volume is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi ^{2}}{2}}r^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{2}}r^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90fcafc44bfb592c66901c51e7c0b9905a73933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.327ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi ^{2}}{2}}r^{4}}"></span>.<sup id="cite_ref-FOOTNOTECoxeter1973119_13-1" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973119-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Volume_of_an_n-ball" title="Volume of an n-ball">volume of an <i>n</i>-ball</a> in an arbitrary dimension <i>n</i> is computable from a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> connecting dimension <span class="texhtml mvar" style="font-style:italic;">n</span> to dimension <span class="texhtml"><i>n</i> - 2</span>. </p> <div class="mw-heading mw-heading2"><h2 id="In_culture">In culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=13" title="Edit section: In culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_art">In art</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=14" title="Edit section: In art"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Fourth_dimension_in_art" title="Fourth dimension in art">Fourth dimension in art</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Fourth_dimension_in_art&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Jouffret.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Jouffret.gif/220px-Jouffret.gif" decoding="async" width="220" height="305" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Jouffret.gif/330px-Jouffret.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3b/Jouffret.gif 2x" data-file-width="399" data-file-height="554" /></a><figcaption>An illustration from Jouffret's <i>Traité élémentaire de géométrie à quatre dimensions</i>. The book, which influenced Picasso, was given to him by Princet.</figcaption></figure> New possibilities opened up by the concept of four-dimensional space (and difficulties involved in trying to visualize it) helped inspire many modern artists in the first half of the twentieth century. Early <a href="/wiki/Cubists" class="mw-redirect" title="Cubists">Cubists</a>, <a href="/wiki/Surrealists" class="mw-redirect" title="Surrealists">Surrealists</a>, <a href="/wiki/Futurists" class="mw-redirect" title="Futurists">Futurists</a>, and <a href="/wiki/Abstract_art" title="Abstract art">abstract</a> artists took ideas from <a href="/wiki/Higher-dimensional_space" class="mw-redirect" title="Higher-dimensional space">higher-dimensional</a> mathematics and used them to radically advance their work.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading3"><h3 id="In_literature">In literature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=15" title="Edit section: In literature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fourth_dimension_in_literature" title="Fourth dimension in literature">Fourth dimension in literature</a></div> <p><a href="/wiki/Science_fiction" title="Science fiction">Science fiction</a> texts often mention the concept of "dimension" when referring to <a href="/wiki/Parallel_universes_in_fiction" title="Parallel universes in fiction">parallel or alternate universes</a> or other imagined <a href="/wiki/Plane_(esotericism)" title="Plane (esotericism)">planes of existence</a>. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones. </p><p>One of the most heralded science fiction stories regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novella <i><a href="/wiki/Flatland" title="Flatland">Flatland</a></i> by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described <i>Flatland</i> as "The best introduction one can find into the manner of perceiving dimensions." </p><p>The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in <a href="/wiki/Miles_J._Breuer" class="mw-redirect" title="Miles J. Breuer">Miles J. Breuer</a>'s <i>The Appendix and the Spectacles</i> (1928) and <a href="/wiki/Murray_Leinster" title="Murray Leinster">Murray Leinster</a>'s <i>The Fifth-Dimension Catapult</i> (1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include <a href="/wiki/Robert_A._Heinlein" title="Robert A. Heinlein">Robert A. Heinlein</a>'s <i><a href="/wiki/%E2%80%94And_He_Built_a_Crooked_House" class="mw-redirect" title="—And He Built a Crooked House">—And He Built a Crooked House</a></i> (1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract; <a href="/wiki/Alan_E._Nourse" title="Alan E. Nourse">Alan E. Nourse</a>'s <i>Tiger by the Tail</i> and <i>The Universe Between</i> (both 1951); and <i><a rel="nofollow" class="external text" href="https://archive.org/stream/galaxymagazine-1957-04/Galaxy_1957_04#page/n59/mode/2up">The Ifth of Oofth</a></i> (1957) by <a href="/wiki/Walter_Tevis" title="Walter Tevis">Walter Tevis</a>. Another reference is <a href="/wiki/Madeleine_L%27Engle" title="Madeleine L'Engle">Madeleine L'Engle</a>'s novel <i><a href="/wiki/A_Wrinkle_In_Time" class="mw-redirect" title="A Wrinkle In Time">A Wrinkle In Time</a></i> (1962), which uses the fifth dimension as a way of "tesseracting the universe" or "folding" space to move across it quickly. The fourth and fifth dimensions are also key components of the book <i><a href="/wiki/The_Boy_Who_Reversed_Himself" title="The Boy Who Reversed Himself">The Boy Who Reversed Himself</a></i> by <a href="/wiki/William_Sleator" title="William Sleator">William Sleator</a>. </p> <div class="mw-heading mw-heading3"><h3 id="In_philosophy">In philosophy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=16" title="Edit section: In philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Immanuel_Kant" title="Immanuel Kant">Immanuel Kant</a> wrote in 1783: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space, in general, cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition <i>a priori</i> because it is apodictically (demonstrably) certain."<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>"Space has Four Dimensions" is a short story published in 1846 by German philosopher and <a href="/wiki/Experimental_psychology" title="Experimental psychology">experimental psychologist</a> <a href="/wiki/Gustav_Fechner" title="Gustav Fechner">Gustav Fechner</a> under the <a href="/wiki/Pseudonym" title="Pseudonym">pseudonym</a> "Dr. Mises". The protagonist in the tale is a shadow who is aware of and able to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The story bears a strong similarity to the "<a href="/wiki/Allegory_of_the_Cave" class="mw-redirect" title="Allegory of the Cave">Allegory of the Cave</a>" presented in <a href="/wiki/Plato" title="Plato">Plato</a>'s <i><a href="/wiki/The_Republic_(Plato)" class="mw-redirect" title="The Republic (Plato)">The Republic</a></i> (<abbr title="circa">c.</abbr> 380 BC). </p><p>Simon Newcomb wrote an article for the <i>Bulletin of the American Mathematical Society</i> in 1898 entitled "The Philosophy of Hyperspace".<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Linda_Dalrymple_Henderson" title="Linda Dalrymple Henderson">Linda Dalrymple Henderson</a> coined the term "hyperspace philosophy", used to describe writing that uses higher dimensions to explore <a href="/wiki/Metaphysics" title="Metaphysics">metaphysical</a> themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Examples of "hyperspace philosophers" include <a href="/wiki/Charles_Howard_Hinton" title="Charles Howard Hinton">Charles Howard Hinton</a>, the first writer, in 1888, to use the word "tesseract";<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and the Russian <a href="/wiki/Esotericism" class="mw-redirect" title="Esotericism">esotericist</a> <a href="/wiki/P._D._Ouspensky" title="P. D. Ouspensky">P. D. Ouspensky</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/4-polytope" title="4-polytope">4-polytope</a></li> <li><a href="/wiki/4-manifold" title="4-manifold">4-manifold</a></li> <li><a href="/wiki/Exotic_R4" title="Exotic R4">Exotic <b>R</b><sup>4</sup></a></li> <li><a href="/wiki/Four-dimensionalism" title="Four-dimensionalism">Four-dimensionalism</a></li> <li><a href="/wiki/List_of_four-dimensional_games" title="List of four-dimensional games">List of four-dimensional games</a></li> <li><a href="/wiki/Time_in_physics" title="Time in physics">Time in physics</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=18" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCajori1926" class="citation journal cs1">Cajori, Florian (1926). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00029890.1926.11986607">"Origins of Fourth Dimension Concepts"</a>. <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>33</b> (8) (published March 6, 2018): 397–406. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1926.11986607">10.1080/00029890.1926.11986607</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a><span class="reference-accessdate">. Retrieved <span class="nowrap">October 10,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Origins+of+Fourth+Dimension+Concepts&rft.volume=33&rft.issue=8&rft.pages=397-406&rft.date=1926&rft_id=info%3Adoi%2F10.1080%2F00029890.1926.11986607&rft.issn=0002-9890&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F00029890.1926.11986607&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESchläfli1901-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESchläfli1901_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchläfli1901">Schläfli 1901</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVesselin_Petkov2007" class="citation book cs1">Vesselin Petkov (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l9jeb2b3QIIC"><i>Relativity and the Dimensionality of the World</i></a> (illustrated ed.). Springer Science & Business Media. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-6317-6" title="Special:BookSources/978-1-4020-6317-6"><bdi>978-1-4020-6317-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativity+and+the+Dimensionality+of+the+World&rft.pages=23&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-1-4020-6317-6&rft.au=Vesselin+Petkov&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dl9jeb2b3QIIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l9jeb2b3QIIC&pg=PA23">Extract of page 23</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1965" class="citation book cs1">Bell, E.T. (1965). <i>Men of Mathematics</i> (1st ed.). New York: <a href="/wiki/Simon_and_Schuster" class="mw-redirect" title="Simon and Schuster">Simon and Schuster</a>. p. 154. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-671-62818-5" title="Special:BookSources/978-0-671-62818-5"><bdi>978-0-671-62818-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Men+of+Mathematics&rft.place=New+York&rft.pages=154&rft.edition=1st&rft.pub=Simon+and+Schuster&rft.date=1965&rft.isbn=978-0-671-62818-5&rft.aulast=Bell&rft.aufirst=E.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTECoxeter1973141§7.x._Historical_remarks-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1973141§7.x._Historical_remarks_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 141, §7.x. Historical remarks; "<a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a> realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by <a href="/wiki/H._G._Wells" title="H. G. Wells">H. G. Wells</a> in <i>The Plattner Story</i>."</span> </li> <li id="cite_note-FOOTNOTECoxeter1973141–144§7._Ordinary_Polytopes_in_Higher_Space;_§7.x._Historical_remarks-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1973141–144§7._Ordinary_Polytopes_in_Higher_Space;_§7.x._Historical_remarks_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 141–144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlegelWaren1886" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Victor_Schlegel" title="Victor Schlegel">Schlegel, Victor</a>; Waren (1886). <i>Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper</i> [<i>On projection models of regular four-dimensional bodies</i>] (in German).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ueber+Projectionsmodelle+der+regelm%C3%A4ssigen+vier-dimensionalen+K%C3%B6rper&rft.date=1886&rft.aulast=Schlegel&rft.aufirst=Victor&rft.au=Waren&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinton1980" class="citation book cs1"><a href="/wiki/Charles_Howard_Hinton" title="Charles Howard Hinton">Hinton, Charles Howard</a> (1980). Rucker, Rudolf v. B. (ed.). <i>Speculations on the Fourth Dimension: Selected writings of Charles H. Hinton</i>. New York: <a href="/w/index.php?title=Dover_Publishing&action=edit&redlink=1" class="new" title="Dover Publishing (page does not exist)">Dover Publishing</a>. p. vii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-23916-3" title="Special:BookSources/978-0-486-23916-3"><bdi>978-0-486-23916-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Speculations+on+the+Fourth+Dimension%3A+Selected+writings+of+Charles+H.+Hinton&rft.place=New+York&rft.pages=vii&rft.pub=Dover+Publishing&rft.date=1980&rft.isbn=978-0-486-23916-3&rft.aulast=Hinton&rft.aufirst=Charles+Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-Hinton-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hinton_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hinton_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hinton_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinton1993" class="citation book cs1">Hinton, Charles Howard (1993) [1904]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_ZG3MA1wvjIC&pg=PA14"><i>The Fourth Dimension</i></a>. Pomeroy, Washington: Health Research. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7873-0410-2" title="Special:BookSources/978-0-7873-0410-2"><bdi>978-0-7873-0410-2</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 February</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fourth+Dimension&rft.place=Pomeroy%2C+Washington&rft.pages=14&rft.pub=Health+Research&rft.date=1993&rft.isbn=978-0-7873-0410-2&rft.aulast=Hinton&rft.aufirst=Charles+Howard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_ZG3MA1wvjIC%26pg%3DPA14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1975" class="citation book cs1">Gardner, Martin (1975). <i>Mathematical Carnival: From Penny Puzzles. Card Shuffles and Tricks of Lightning Calculators to Roller Coaster Rides into the Fourth Dimension</i> (1st ed.). New York: <a href="/wiki/Knopf_Publishing" class="mw-redirect" title="Knopf Publishing">Knopf</a>. pp. 42, 52–53. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-394-49406-7" title="Special:BookSources/978-0-394-49406-7"><bdi>978-0-394-49406-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Carnival%3A+From+Penny+Puzzles.+Card+Shuffles+and+Tricks+of+Lightning+Calculators+to+Roller+Coaster+Rides+into+the+Fourth+Dimension&rft.place=New+York&rft.pages=42%2C+52-53&rft.edition=1st&rft.pub=Knopf&rft.date=1975&rft.isbn=978-0-394-49406-7&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMinkowski1909" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski, Hermann</a> (1909). <a class="external text" href="https://en.wikisource.org/wiki/de:Raum_und_Zeit_(Minkowski)">"Raum und Zeit"</a> [Space and Time]. <i>Physikalische Zeitschrift</i> (in German). <b>10</b>: 75–88<span class="reference-accessdate">. Retrieved <span class="nowrap">October 27,</span> 2022</span> – via Wikisource.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift&rft.atitle=Raum+und+Zeit&rft.volume=10&rft.pages=75-88&rft.date=1909&rft.aulast=Minkowski&rft.aufirst=Hermann&rft_id=https%3A%2F%2Fen.wikisource.org%2Fwiki%2Fde%3ARaum_und_Zeit_%28Minkowski%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-Møller-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Møller_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMøller1972" class="citation book cs1">Møller, C. (1972). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryofrelativi0000mlle"><i>The Theory of Relativity</i></a></span> (2nd ed.). Oxford: Clarendon Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryofrelativi0000mlle/page/93">93</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851256-1" title="Special:BookSources/978-0-19-851256-1"><bdi>978-0-19-851256-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Relativity&rft.place=Oxford&rft.pages=93&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1972&rft.isbn=978-0-19-851256-1&rft.aulast=M%C3%B8ller&rft.aufirst=C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryofrelativi0000mlle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTECoxeter1973119-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTECoxeter1973119_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTECoxeter1973119_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 119.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarterSaito" class="citation book cs1">Carter, J. Scott; Saito, Masahico. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TIGVq4GeEM4C"><i>Knotted Surfaces and Their Diagrams</i></a>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-7491-2" title="Special:BookSources/978-0-8218-7491-2"><bdi>978-0-8218-7491-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knotted+Surfaces+and+Their+Diagrams&rft.pub=American+Mathematical+Society&rft.isbn=978-0-8218-7491-2&rft.aulast=Carter&rft.aufirst=J.+Scott&rft.au=Saito%2C+Masahico&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTIGVq4GeEM4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD'Inverno1998" class="citation book cs1">D'Inverno, Ray (1998). <i>Introducing Einstein's Relativity</i> (Reprint ed.). 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(2008). <a rel="nofollow" class="external text" href="https://grazianolab.princeton.edu/sites/default/files/graziano/files/aflalo_08.pdf">"Four-dimensional spatial reasoning in humans"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Experimental Psychology: Human Perception and Performance</i>. <b>34</b> (5): 1066–1077. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.505.5736">10.1.1.505.5736</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0096-1523.34.5.1066">10.1037/0096-1523.34.5.1066</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18823195">18823195</a><span class="reference-accessdate">. Retrieved <span class="nowrap">20 August</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Experimental+Psychology%3A+Human+Perception+and+Performance&rft.atitle=Four-dimensional+spatial+reasoning+in+humans&rft.volume=34&rft.issue=5&rft.pages=1066-1077&rft.date=2008&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.505.5736%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F18823195&rft_id=info%3Adoi%2F10.1037%2F0096-1523.34.5.1066&rft.aulast=Aflalo&rft.aufirst=T.+N.&rft.au=Graziano%2C+M.+S.+A.&rft_id=https%3A%2F%2Fgrazianolab.princeton.edu%2Fsites%2Fdefault%2Ffiles%2Fgraziano%2Ffiles%2Faflalo_08.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-McIntosh-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-McIntosh_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcIntosh2002" class="citation web cs1">McIntosh, John (November 2002). <a rel="nofollow" class="external text" href="https://www.urticator.net/maze/">"4D Maze Game"</a>. <i>urticator.net</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-12-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=urticator.net&rft.atitle=4D+Maze+Game&rft.date=2002-11&rft.aulast=McIntosh&rft.aufirst=John&rft_id=https%3A%2F%2Fwww.urticator.net%2Fmaze%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-Ogmen-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ogmen_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOgmenShibataYazdanbakhsh2020" class="citation journal cs1">Ogmen, Haluk; Shibata, Kazuhisa; Yazdanbakhsh, Arash (2020-01-22). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6987450">"Perception, Cognition, and Action in Hyperspaces: Implications on Brain Plasticity, Learning, and Cognition"</a>. <i><a href="/wiki/Frontiers_in_Psychology" title="Frontiers in Psychology">Frontiers in Psychology</a></i>. <b>10</b>. <a href="/wiki/Frontiers_Media" title="Frontiers Media">Frontiers Media</a>: 3000. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3389%2Ffpsyg.2019.03000">10.3389/fpsyg.2019.03000</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6987450">6987450</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32038384">32038384</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Frontiers+in+Psychology&rft.atitle=Perception%2C+Cognition%2C+and+Action+in+Hyperspaces%3A+Implications+on+Brain+Plasticity%2C+Learning%2C+and+Cognition&rft.volume=10&rft.pages=3000&rft.date=2020-01-22&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6987450%23id-name%3DPMC&rft_id=info%3Apmid%2F32038384&rft_id=info%3Adoi%2F10.3389%2Ffpsyg.2019.03000&rft.aulast=Ogmen&rft.aufirst=Haluk&rft.au=Shibata%2C+Kazuhisa&rft.au=Yazdanbakhsh%2C+Arash&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6987450&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaku1995" class="citation book cs1"><a href="/wiki/Michio_Kaku" title="Michio Kaku">Kaku, Michio</a> (1995). <a href="/wiki/Hyperspace_(book)" title="Hyperspace (book)"><i>Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension</i></a> (reissued ed.). Oxford: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. pp. Part I, Chapter 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-286189-4" title="Special:BookSources/978-0-19-286189-4"><bdi>978-0-19-286189-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hyperspace%3A+A+Scientific+Odyssey+Through+Parallel+Universes%2C+Time+Warps%2C+and+the+Tenth+Dimension&rft.place=Oxford&rft.pages=Part+I%2C+Chapter+3&rft.edition=reissued&rft.pub=Oxford+University+Press&rft.date=1995&rft.isbn=978-0-19-286189-4&rft.aulast=Kaku&rft.aufirst=Michio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRucker1996" class="citation book cs1">Rucker, Rudy (1996). <i>The Fourth Dimension: A Guided Tour of the Higher Universe</i>. 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Retrieved <span class="nowrap">24 March</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Overview+of+The+Fourth+Dimension+And+Non-Euclidean+Geometry+In+Modern+Art%2C+Revised+Edition&rft.pub=MIT+Press&rft.aulast=Henderson&rft.aufirst=Linda+Dalrymple&rft_id=http%3A%2F%2Fmitpress.mit.edu%2Fbooks%2Ffourth-dimension-and-non-euclidean-geometry-modern-art&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><i><a href="/wiki/Prolegomena_to_Any_Future_Metaphysics_That_Will_Be_Able_to_Present_Itself_as_a_Science" class="mw-redirect" title="Prolegomena to Any Future Metaphysics That Will Be Able to Present Itself as a Science">Prolegomena</a></i>, § 12</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBanchoff1990" class="citation journal cs1">Banchoff, Thomas F. (1990). <a rel="nofollow" class="external text" href="http://www.geom.uiuc.edu/~banchoff/ISR/ISR.html">"From Flatland to Hypergraphics: Interacting with Higher Dimensions"</a>. <i>Interdisciplinary Science Reviews</i>. <b>15</b> (4): 364. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990ISRv...15..364B">1990ISRv...15..364B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1179%2F030801890789797239">10.1179/030801890789797239</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130414023125/http://www.geom.uiuc.edu/~banchoff/ISR/ISR.html">Archived</a> from the original on 2013-04-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Interdisciplinary+Science+Reviews&rft.atitle=From+Flatland+to+Hypergraphics%3A+Interacting+with+Higher+Dimensions&rft.volume=15&rft.issue=4&rft.pages=364&rft.date=1990&rft_id=info%3Adoi%2F10.1179%2F030801890789797239&rft_id=info%3Abibcode%2F1990ISRv...15..364B&rft.aulast=Banchoff&rft.aufirst=Thomas+F.&rft_id=http%3A%2F%2Fwww.geom.uiuc.edu%2F~banchoff%2FISR%2FISR.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewcomb1898" class="citation journal cs1">Newcomb, Simon (1898). <a rel="nofollow" class="external text" href="https://archive.org/details/cihm_42903">"The Philosophy of Hyperspace"</a>. <i>Bulletin of the American Mathematical Society</i>. <b>4</b> (5): 187. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1898-00478-0">10.1090/S0002-9904-1898-00478-0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=The+Philosophy+of+Hyperspace&rft.volume=4&rft.issue=5&rft.pages=187&rft.date=1898&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1898-00478-0&rft.aulast=Newcomb&rft.aufirst=Simon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcihm_42903&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKruger2007" class="citation journal cs1">Kruger, Runette (2007). <a rel="nofollow" class="external text" href="http://ler.letras.up.pt/uploads/ficheiros/4351.pdf">"Art in the Fourth Dimension: Giving Form to Form – The Abstract Paintings of Piet Mondrian"</a> <span class="cs1-format">(PDF)</span>. <i>Spaces of Utopia: An Electronic Journal</i> (5): 11. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110929071218/http://ler.letras.up.pt/uploads/ficheiros/4351.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2011-09-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Spaces+of+Utopia%3A+An+Electronic+Journal&rft.atitle=Art+in+the+Fourth+Dimension%3A+Giving+Form+to+Form+%E2%80%93+The+Abstract+Paintings+of+Piet+Mondrian&rft.issue=5&rft.pages=11&rft.date=2007&rft.aulast=Kruger&rft.aufirst=Runette&rft_id=http%3A%2F%2Fler.letras.up.pt%2Fuploads%2Fficheiros%2F4351.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickover2009" class="citation cs2"><a href="/wiki/Clifford_Pickover" class="mw-redirect" title="Clifford Pickover">Pickover, Clifford A.</a> (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JrslMKTgSZwC&pg=PA282">"Tesseract"</a>, <i>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics</i>, Sterling Publishing, p. 282, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4027-5796-9" title="Special:BookSources/978-1-4027-5796-9"><bdi>978-1-4027-5796-9</bdi></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170330181025/https://books.google.com/books?id=JrslMKTgSZwC&pg=PA282">archived</a> from the original on 2017-03-30</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tesseract&rft.btitle=The+Math+Book%3A+From+Pythagoras+to+the+57th+Dimension%2C+250+Milestones+in+the+History+of+Mathematics&rft.pages=282&rft.pub=Sterling+Publishing&rft.date=2009&rft.isbn=978-1-4027-5796-9&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJrslMKTgSZwC%26pg%3DPA282&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchläfli1901" class="citation cs2 cs1-prop-foreign-lang-source">Schläfli, Ludwig (1901) [1852], Graf, J. H. (ed.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=foIUAQAAMAAJ"><i>Theorie der vielfachen Kontinuität</i></a>, Republished by Cornell University Library historical math monographs 2010 (in German), Zürich, Basel: Georg & Co., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4297-0481-6" title="Special:BookSources/978-1-4297-0481-6"><bdi>978-1-4297-0481-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+der+vielfachen+Kontinuit%C3%A4t&rft.series=Republished+by+Cornell+University+Library+historical+math+monographs+2010&rft.pub=Z%C3%BCrich%2C+Basel%3A+Georg+%26+Co.&rft.date=1901&rft.isbn=978-1-4297-0481-6&rft.aulast=Schl%C3%A4fli&rft.aufirst=Ludwig&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfoIUAQAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1973" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1973) [1948]. <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)"><i>Regular Polytopes</i></a> (3rd ed.). New York: Dover.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Polytopes&rft.place=New+York&rft.edition=3rd&rft.pub=Dover&rft.date=1973&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=20" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArchibald1914" class="citation journal cs1"><a href="/wiki/Raymond_Clare_Archibald" class="mw-redirect" title="Raymond Clare Archibald">Archibald, R. C</a> (1914). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1914-20-08/S0002-9904-1914-02511-X/S0002-9904-1914-02511-X.pdf">"Time as a Fourth Dimension"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the American Mathematical Society</i>: 409–412.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Time+as+a+Fourth+Dimension&rft.pages=409-412&rft.date=1914&rft.aulast=Archibald&rft.aufirst=R.+C&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1914-20-08%2FS0002-9904-1914-02511-X%2FS0002-9904-1914-02511-X.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span></li> <li><a href="/wiki/Andrew_Forsyth" title="Andrew Forsyth">Andrew Forsyth</a> (1930) <a rel="nofollow" class="external text" href="https://archive.org/details/geometryoffourdi032760mbp">Geometry of Four Dimensions</a>, link from <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGamow1988" class="citation book cs1"><a href="/wiki/George_Gamow" title="George Gamow">Gamow, George</a> (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EZbcwk6SkhcC"><i>One Two Three . . . Infinity: Facts and Speculations of Science</i></a> (3rd ed.). Courier Dover Publications. p. 68. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-25664-1" title="Special:BookSources/978-0-486-25664-1"><bdi>978-0-486-25664-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=One+Two+Three+.+.+.+Infinity%3A+Facts+and+Speculations+of+Science&rft.pages=68&rft.edition=3rd&rft.pub=Courier+Dover+Publications&rft.date=1988&rft.isbn=978-0-486-25664-1&rft.aulast=Gamow&rft.aufirst=George&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEZbcwk6SkhcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFour-dimensional+space" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EZbcwk6SkhcC&pg=PA68">Extract of page 68</a></li> <li><a href="/wiki/E._H._Neville" class="mw-redirect" title="E. H. Neville">E. H. Neville</a> (1921) <a rel="nofollow" class="external text" href="http://quod.lib.umich.edu/u/umhistmath/ABR2619.0001.001?rgn=works;view=toc;rgn1=author;q1=Neville"><i>The Fourth Dimension</i></a>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, link from <a href="/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> Historical Math Collection.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Four-dimensional_space&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Special_Relativity" class="extiw" title="wikibooks:Special Relativity">Special Relativity</a></b></i></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.dimensions-math.org">"Dimensions" videos, showing several different ways to visualize four-dimensional objects</a></li> <li><a rel="nofollow" class="external text" href="http://www.sciencenews.org/index/generic/activity/view/id/35740/title/Math_Trek__Seeing_in_four_dimensions"><i>Science News</i> article summarizing the "Dimensions" videos, with clips</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120929062930/http://www.sciencenews.org/index/generic/activity/view/id/35740/title/Math_Trek__Seeing_in_four_dimensions">Archived</a> 2012-09-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a href="https://en.wikisource.org/wiki/Flatland_(second_edition)" class="extiw" title="s:Flatland (second edition)"><i>Flatland: a Romance of Many Dimensions</i> (second edition)</a></li> <li><a rel="nofollow" class="external text" href="http://www.math.union.edu/~dpvc/math/4D/welcome.html">Frame-by-frame animations of 4D - 3D analogies</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Dimension" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Dimension_topics" title="Template:Dimension topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Dimension_topics" title="Template talk:Dimension topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Dimension_topics" title="Special:EditPage/Template:Dimension topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Dimension" style="font-size:114%;margin:0 4em"><a href="/wiki/Dimension" title="Dimension">Dimension</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensional spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">Vector space</a></li> <li><a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine space</a></li> <li><a href="/wiki/Projective_space" title="Projective space">Projective space</a></li> <li><a href="/wiki/Free_module" title="Free module">Free module</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Tesseract.gif" class="mw-file-description" title="Animated tesseract"><img alt="Animated tesseract" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/75px-Tesseract.gif" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/113px-Tesseract.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/150px-Tesseract.gif 2x" data-file-width="256" data-file-height="256" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other dimensions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Krull_dimension" title="Krull dimension">Krull</a></li> <li><a href="/wiki/Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering</a></li> <li><a href="/wiki/Inductive_dimension" title="Inductive dimension">Inductive</a></li> <li><a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li> <li><a href="/wiki/Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Minkowski</a></li> <li><a href="/wiki/Fractal_dimension" title="Fractal dimension">Fractal</a></li> <li><a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">Degrees of freedom</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polytope" title="Polytope">Polytopes</a> and <a href="/wiki/Shape" title="Shape">shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hyperplane" title="Hyperplane">Hyperplane</a></li> <li><a href="/wiki/Hypersurface" title="Hypersurface">Hypersurface</a></li> <li><a href="/wiki/Hypercube" title="Hypercube">Hypercube</a></li> <li><a href="/wiki/Hyperrectangle" title="Hyperrectangle">Hyperrectangle</a></li> <li><a href="/wiki/Demihypercube" title="Demihypercube">Demihypercube</a></li> <li><a href="/wiki/N-sphere" title="N-sphere">Hypersphere</a></li> <li><a href="/wiki/Cross-polytope" title="Cross-polytope">Cross-polytope</a></li> <li><a href="/wiki/Simplex" title="Simplex">Simplex</a></li> <li><a href="/wiki/Hyperpyramid" title="Hyperpyramid">Hyperpyramid</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Number systems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hypercomplex_number" title="Hypercomplex number">Hypercomplex numbers</a></li> <li><a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensions by number</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero</a></li> <li><a href="/wiki/One-dimensional_space" title="One-dimensional space">One</a></li> <li><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two</a></li> <li><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three</a></li> <li><a class="mw-selflink selflink">Four</a></li> <li><a href="/wiki/Five-dimensional_space" title="Five-dimensional space">Five</a></li> <li><a href="/wiki/Six-dimensional_space" title="Six-dimensional space">Six</a></li> <li><a href="/wiki/Seven-dimensional_space" title="Seven-dimensional space">Seven</a></li> <li><a href="/wiki/Eight-dimensional_space" title="Eight-dimensional space">Eight</a></li> <li><a href="/wiki/Dimension" title="Dimension"><i>n</i>-dimensions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hyperspace" title="Hyperspace">Hyperspace</a></li> <li><a href="/wiki/Codimension" title="Codimension">Codimension</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b><a href="/wiki/Category:Dimension" title="Category:Dimension">Category</a></b></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Four-dimensional_space&oldid=1257030709">https://en.wikipedia.org/w/index.php?title=Four-dimensional_space&oldid=1257030709</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" 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