Simplex - Wikipedia

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class="vector-toc-link" href="#Symmetric_graphs_of_regular_simplices"> <div class="vector-toc-text"> 3 Symmetric graphs of regular simplices </div> </a> <ul id="toc-Symmetric_graphs_of_regular_simplices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Standard_simplex" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Standard_simplex"> <div class="vector-toc-text"> 4 Standard simplex </div> </a> <button aria-controls="toc-Standard_simplex-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Standard simplex subsection </button> <ul id="toc-Standard_simplex-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 4.1 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Increasing_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Increasing_coordinates"> <div class="vector-toc-text"> 4.2 Increasing coordinates </div> </a> <ul id="toc-Increasing_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projection_onto_the_standard_simplex" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projection_onto_the_standard_simplex"> <div class="vector-toc-text"> 4.3 Projection onto the standard simplex </div> </a> <ul id="toc-Projection_onto_the_standard_simplex-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corner_of_cube" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Corner_of_cube"> <div class="vector-toc-text"> 4.4 Corner of cube </div> </a> <ul id="toc-Corner_of_cube-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cartesian_coordinates_for_a_regular_n-dimensional_simplex_in_Rn" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cartesian_coordinates_for_a_regular_n-dimensional_simplex_in_Rn"> <div class="vector-toc-text"> 5 Cartesian coordinates for a regular n-dimensional simplex in Rn </div> </a> <ul id="toc-Cartesian_coordinates_for_a_regular_n-dimensional_simplex_in_Rn-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometric_properties"> <div class="vector-toc-text"> 6 Geometric properties </div> </a> <button aria-controls="toc-Geometric_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Geometric properties subsection </button> <ul id="toc-Geometric_properties-sublist" class="vector-toc-list"> <li id="toc-Volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volume"> <div class="vector-toc-text"> 6.1 Volume </div> </a> <ul id="toc-Volume-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dihedral_angles_of_the_regular_n-simplex" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dihedral_angles_of_the_regular_n-simplex"> <div class="vector-toc-text"> 6.2 Dihedral angles of the regular n-simplex </div> </a> <ul id="toc-Dihedral_angles_of_the_regular_n-simplex-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simplices_with_an_"orthogonal_corner"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simplices_with_an_"orthogonal_corner""> <div class="vector-toc-text"> 6.3 Simplices with an "orthogonal corner" </div> </a> <ul id="toc-Simplices_with_an_"orthogonal_corner"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_(n_+_1)-hypercube" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_(n_+_1)-hypercube"> <div class="vector-toc-text"> 6.4 Relation to the (n + 1)-hypercube </div> </a> <ul id="toc-Relation_to_the_(n_+_1)-hypercube-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topology"> <div class="vector-toc-text"> 6.5 Topology </div> </a> <ul id="toc-Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability"> <div class="vector-toc-text"> 6.6 Probability </div> </a> <ul id="toc-Probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aitchison_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aitchison_geometry"> <div class="vector-toc-text"> 6.7 Aitchison geometry </div> </a> <ul id="toc-Aitchison_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compounds"> <div class="vector-toc-text"> 6.8 Compounds </div> </a> <ul id="toc-Compounds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebraic_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_topology"> <div class="vector-toc-text"> 7 Algebraic topology </div> </a> <ul id="toc-Algebraic_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_geometry"> <div class="vector-toc-text"> 8 Algebraic geometry </div> </a> <ul id="toc-Algebraic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 9 Applications </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </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"> 10 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 11 Notes </div> </a> <ul id="toc-Notes-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"> 12 References </div> </a> <ul id="toc-References-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" title="Table of Contents" > <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" > Toggle the table of contents </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">Simplex</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 34 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-34" aria-hidden="true" > 34 languages </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%85%D8%A8%D8%B3%D8%B7" title="مبسط – Arabic" lang="ar" hreflang="ar" data-title="مبسط" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%AE%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A7%80_%E0%A6%B8%E0%A6%BF%E0%A6%AE%E0%A6%AA%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B8" title="ক্রমবর্তী সিমপ্লেক্স – Bangla" lang="bn" hreflang="bn" data-title="ক্রমবর্তী সিমপ্লেক্স" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплекс – Bulgarian" lang="bg" hreflang="bg" data-title="Симплекс" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/S%C3%ADmplex" title="Símplex – Catalan" lang="ca" hreflang="ca" data-title="Símplex" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплекс – Chuvash" lang="cv" hreflang="cv" data-title="Симплекс" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Simplex" title="Simplex – Czech" lang="cs" hreflang="cs" data-title="Simplex" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Simplex_(Mathematik)" title="Simplex (Mathematik) – German" lang="de" hreflang="de" data-title="Simplex (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/S%C3%ADmplex" title="Símplex – Spanish" lang="es" hreflang="es" data-title="Símplex" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Simpla%C4%B5o_(geometrio)" title="Simplaĵo (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Simplaĵo (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Simplex_(geometria)" title="Simplex (geometria) – Basque" lang="eu" hreflang="eu" data-title="Simplex (geometria)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D8%A7%D8%AF%DA%A9" title="سادک – Persian" lang="fa" hreflang="fa" data-title="سادک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Simplexe" title="Simplexe – French" lang="fr" hreflang="fr" data-title="Simplexe" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A8%EC%B2%B4_(%EC%88%98%ED%95%99)" title="단체 (수학) – Korean" lang="ko" hreflang="ko" data-title="단체 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%AB%D5%B4%D5%BA%D5%AC%D5%A5%D6%84%D5%BD" title="Սիմպլեքս – Armenian" lang="hy" hreflang="hy" data-title="Սիմպլեքս" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Simplesso" title="Simplesso – Italian" lang="it" hreflang="it" data-title="Simplesso" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%A4%D7%9C%D7%A7%D7%A1" title="סימפלקס – Hebrew" lang="he" hreflang="he" data-title="סימפלקס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплекс – Kazakh" lang="kk" hreflang="kk" data-title="Симплекс" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szimplex" title="Szimplex – Hungarian" lang="hu" hreflang="hu" data-title="Szimplex" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Simplex_(wiskunde)" title="Simplex (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Simplex (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%98%E4%BD%93_(%E6%95%B0%E5%AD%A6)" title="単体 (数学) – Japanese" lang="ja" hreflang="ja" data-title="単体 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Simpleks_(matematikk)" title="Simpleks (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Simpleks (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Simpleks" title="Simpleks – Uzbek" lang="uz" hreflang="uz" data-title="Simpleks" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sympleks_(matematyka)" title="Sympleks (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Sympleks (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Simplex_(topologia)" title="Simplex (topologia) – Portuguese" lang="pt" hreflang="pt" data-title="Simplex (topologia)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Simplex" title="Simplex – Romanian" lang="ro" hreflang="ro" data-title="Simplex" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплекс – Russian" lang="ru" hreflang="ru" data-title="Симплекс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Simpleks" title="Simpleks – Albanian" lang="sq" hreflang="sq" data-title="Simpleks" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Simplex" title="Simplex – Simple English" lang="en-simple" hreflang="en-simple" data-title="Simplex" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Simplex_(geometria)" title="Simplex (geometria) – Slovak" lang="sk" hreflang="sk" data-title="Simplex (geometria)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Simpleks" title="Simpleks – Slovenian" lang="sl" hreflang="sl" data-title="Simpleks" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Simplex" title="Simplex – Swedish" lang="sv" hreflang="sv" data-title="Simplex" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%8B%E0%B8%B4%E0%B8%A1%E0%B9%80%E0%B8%9E%E0%B8%A5%E0%B9%87%E0%B8%81%E0%B8%8B%E0%B9%8C" title="ซิมเพล็กซ์ – Thai" lang="th" hreflang="th" data-title="ซิมเพล็กซ์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81" title="Симплекс – Ukrainian" lang="uk" hreflang="uk" data-title="Симплекс" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%95%E7%BA%AF%E5%BD%A2" title="单纯形 – Chinese" lang="zh" hreflang="zh" data-title="单纯形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q331350#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">For other uses, see <a href="/wiki/Simplex_(disambiguation)" class="mw-disambig" title="Simplex (disambiguation)">Simplex (disambiguation)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Multi-dimensional generalization of triangle</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Simplexes.jpg" class="mw-file-description"><img alt="The four simplexes that can be fully represented in 3D space." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Simplexes.jpg/271px-Simplexes.jpg" decoding="async" width="271" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Simplexes.jpg/407px-Simplexes.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Simplexes.jpg/542px-Simplexes.jpg 2x" data-file-width="1920" data-file-height="1080" /></a><figcaption>The four simplexes that can be fully represented in 3D space.</figcaption></figure> In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a simplex (plural: simplexes or simplices) is a generalization of the notion of a <a href="/wiki/Triangle" title="Triangle">triangle</a> or <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> to arbitrary <a href="/wiki/Dimensions" class="mw-redirect" title="Dimensions">dimensions</a>. The simplex is so-named because it represents the simplest possible <a href="/wiki/Polytope" title="Polytope">polytope</a> in any given dimension. For example, <ul><li>a <a href="/wiki/0-dimensional" class="mw-redirect" title="0-dimensional">0-dimensional</a> simplex is a <a href="/wiki/Point_(mathematics)" class="mw-redirect" title="Point (mathematics)">point</a>,</li> <li>a <a href="/wiki/1-dimensional" class="mw-redirect" title="1-dimensional">1-dimensional</a> simplex is a <a href="/wiki/Line_segment" title="Line segment">line segment</a>,</li> <li>a <a href="/wiki/2-dimensional" class="mw-redirect" title="2-dimensional">2-dimensional</a> simplex is a <a href="/wiki/Triangle" title="Triangle">triangle</a>,</li> <li>a <a href="/wiki/3-dimensional" class="mw-redirect" title="3-dimensional">3-dimensional</a> simplex is a <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, and</li> <li>a <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a> simplex is a <a href="/wiki/5-cell" title="5-cell">5-cell</a>.</li></ul> Specifically, a k-simplex is a k-dimensional <a href="/wiki/Polytope" title="Polytope">polytope</a> that is the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of its k + 1 <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>. More formally, suppose the k + 1 points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{0},\dots ,u_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{0},\dots ,u_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc0aadc0efaea8757b807051395c2689e0505ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.981ex; height:2.009ex;" alt="{\displaystyle u_{0},\dots ,u_{k}}" /> are <a href="/wiki/Affinely_independent" class="mw-redirect" title="Affinely independent">affinely independent</a>, which means that the k vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5252ccba9ce740e41b79e8dcf796ea3a77017e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.429ex; height:2.343ex;" alt="{\displaystyle u_{1}-u_{0},\dots ,u_{k}-u_{0}}" /> are <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a>. Then, the simplex determined by them is the set of points <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7418b84ac0ecc9a880878e90101b6d4a50a7465" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.724ex; height:7.509ex;" alt="{\displaystyle C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.}" /> A regular simplex<a href="#cite_note-1">[1]</a> is a simplex that is also a <a href="/wiki/Regular_polytope" title="Regular polytope">regular polytope</a>. A regular k-simplex may be constructed from a regular (k − 1)-simplex by connecting a new vertex to all original vertices by the common edge length. The standard simplex or probability simplex<a href="#cite_note-Boyd-2">[2]</a> is the (k − 1)-dimensional simplex whose vertices are the k standard <a href="/wiki/Unit_vectors" class="mw-redirect" title="Unit vectors">unit vectors</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11a8204fdaad8aa355c7b0ce28dfa2db9e18c984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.092ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{k}}" />, or in other words <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{x\in \mathbf {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{x\in \mathbf {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbc9e96e4007f7ff0c3c44b0194f7eaf2bf1af2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.339ex; height:3.343ex;" alt="{\displaystyle \left\{x\in \mathbf {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.}" /> In <a href="/wiki/Topology" title="Topology">topology</a> and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, it is common to "glue together" simplices to form a <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a>. The geometric simplex and simplicial complex should not be confused with the <a href="/wiki/Abstract_simplicial_complex" title="Abstract simplicial complex">abstract simplicial complex</a>, in which a simplex is simply a <a href="/wiki/Finite_set" title="Finite set">finite set</a> and the complex is a family of such sets that is closed under taking subsets. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=1" title="Edit section: History">edit</a>]</div> The concept of a simplex was known to <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">William Kingdon Clifford</a>, who wrote about these shapes in 1886 but called them "prime confines". <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>, writing about <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> in 1900, called them "generalized tetrahedra". In 1902 <a href="/wiki/Pieter_Hendrik_Schoute" title="Pieter Hendrik Schoute">Pieter Hendrik Schoute</a> described the concept first with the <a href="/wiki/Latin" title="Latin">Latin</a> superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").<a href="#cite_note-3">[3]</a> The regular simplex family is the first of three <a href="/wiki/Regular_polytope" title="Regular polytope">regular polytope</a> families, labeled by <a href="/wiki/Donald_Coxeter" class="mw-redirect" title="Donald Coxeter">Donald Coxeter</a> as αn, the other two being the <a href="/wiki/Cross-polytope" title="Cross-polytope">cross-polytope</a> family, labeled as βn, and the <a href="/wiki/Hypercube" title="Hypercube">hypercubes</a>, labeled as γn. A fourth family, the <a href="/wiki/Hypercubic_honeycomb" title="Hypercubic honeycomb">tessellation of n-dimensional space by infinitely many hypercubes</a>, he labeled as δn.<a href="#cite_note-FOOTNOTECoxeter1973120–124§7.2-4">[4]</a> <div class="mw-heading mw-heading2"><h2 id="Elements">Elements</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=2" title="Edit section: Elements">edit</a>]</div> The <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of any <a href="/wiki/Empty_set" title="Empty set">nonempty</a> <a href="/wiki/Subset" title="Subset">subset</a> of the n + 1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 (of the n + 1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n+1}{m+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n+1}{m+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4601f68f71945c0d066ae0d2af4cde7d8717347f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.673ex; height:3.509ex;" alt="{\displaystyle {\tbinom {n+1}{m+1}}}" />.<a href="#cite_note-FOOTNOTECoxeter1973120-5">[5]</a> Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a <a href="/wiki/Simplicial_complex#Definitions" title="Simplicial complex">simplicial complex</a>. The extended <a href="/wiki/F-vector" class="mw-redirect" title="F-vector">f-vector</a> for an n-simplex can be computed by (1,1)n+1, like the coefficients of <a href="/wiki/Polynomial#Multiplication" title="Polynomial">polynomial products</a>. For example, a <a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> is (1,1)8 = (1,2,1)4 = (1,4,6,4,1)2 = (1,8,28,56,70,56,28,8,1). The number of 1-faces (edges) of the n-simplex is the n-th <a href="/wiki/Triangle_number" class="mw-redirect" title="Triangle number">triangle number</a>, the number of 2-faces of the n-simplex is the (n − 1)th <a href="/wiki/Tetrahedron_number" class="mw-redirect" title="Tetrahedron number">tetrahedron number</a>, the number of 3-faces of the n-simplex is the (n − 2)th 5-cell number, and so on. <table class="wikitable"> <caption>n-Simplex elements<a href="#cite_note-6">[6]</a> </caption> <tbody><tr> <th>Δn </th> <th>Name </th> <th><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli</a> <a href="/wiki/Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter</a> </th> <th>0- faces (vertices) </th> <th>1- faces (edges) </th> <th>2- faces (faces) </th> <th>3- faces (cells) </th> <th>4- faces   </th> <th>5- faces   </th> <th>6- faces   </th> <th>7- faces   </th> <th>8- faces   </th> <th>9- faces   </th> <th>10- faces   </th> <th>Sum = 2n+1 − 1 </th></tr> <tr> <th>Δ0 </th> <td>0-simplex (<a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">point</a>) </td> <td>( ) <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" /> </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>1 </td></tr> <tr> <th>Δ1 </th> <td>1-simplex (<a href="/wiki/Edge_(geometry)" title="Edge (geometry)">line segment</a>) </td> <td>{ } = ( ) ∨ ( ) = 2⋅( ) <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" /> </td> <td>2 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>3 </td></tr> <tr> <th>Δ2 </th> <td>2-simplex (<a href="/wiki/Triangle" title="Triangle">triangle</a>) </td> <td>{3} = 3⋅( ) <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" /><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" /><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" /> </td> <td>3 </td> <td>3 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>7 </td></tr> <tr> <th>Δ3 </th> <td>3-simplex (<a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>) </td> <td>{3,3} = 4⋅( ) <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" /><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" /><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" /><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" /><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" /> </td> <td>4 </td> <td>6 </td> <td>4 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>15 </td></tr> <tr> <th>Δ4 </th> <td>4-simplex (<a href="/wiki/5-cell" title="5-cell">5-cell</a>) </td> <td>{33} = 5⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>5 </td> <td>10 </td> <td>10 </td> <td>5 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>31 </td></tr> <tr> <th>Δ5 </th> <td><a href="/wiki/5-simplex" title="5-simplex">5-simplex</a> </td> <td>{34} = 6⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>6 </td> <td>15 </td> <td>20 </td> <td>15 </td> <td>6 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>63 </td></tr> <tr> <th>Δ6 </th> <td><a href="/wiki/6-simplex" title="6-simplex">6-simplex</a> </td> <td>{35} = 7⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>7 </td> <td>21 </td> <td>35 </td> <td>35 </td> <td>21 </td> <td>7 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>  </td> <td>127 </td></tr> <tr> <th>Δ7 </th> <td><a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> </td> <td>{36} = 8⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>8 </td> <td>28 </td> <td>56 </td> <td>70 </td> <td>56 </td> <td>28 </td> <td>8 </td> <td>1 </td> <td>  </td> <td>  </td> <td>  </td> <td>255 </td></tr> <tr> <th>Δ8 </th> <td><a href="/wiki/8-simplex" title="8-simplex">8-simplex</a> </td> <td>{37} = 9⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>9 </td> <td>36 </td> <td>84 </td> <td>126 </td> <td>126 </td> <td>84 </td> <td>36 </td> <td>9 </td> <td>1 </td> <td>  </td> <td>  </td> <td>511 </td></tr> <tr> <th>Δ9 </th> <td><a href="/wiki/9-simplex" title="9-simplex">9-simplex</a> </td> <td>{38} = 10⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>10 </td> <td>45 </td> <td>120 </td> <td>210 </td> <td>252 </td> <td>210 </td> <td>120 </td> <td>45 </td> <td>10 </td> <td>1 </td> <td>  </td> <td>1023 </td></tr> <tr> <th>Δ10 </th> <td><a href="/wiki/10-simplex" title="10-simplex">10-simplex</a> </td> <td>{39} = 11⋅( ) <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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>11 </td> <td>55 </td> <td>165 </td> <td>330 </td> <td>462 </td> <td>462 </td> <td>330 </td> <td>165 </td> <td>55 </td> <td>11 </td> <td>1 </td> <td>2047 </td></tr></tbody></table> An n-simplex is the <a href="/wiki/Polytope" title="Polytope">polytope</a> with the fewest vertices that requires n dimensions. Consider a line segment AB as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point E somewhere outside the 3-space. The new shape ABCDE, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space. More formally, an (n + 1)-simplex can be constructed as a join (∨ operator) of an n-simplex and a point, ( ). An (m + n + 1)-simplex can be constructed as a join of an m-simplex and an n-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangle</a> is the join of a 1-simplex and a point: { } ∨ ( ). An <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A <a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedron</a> is 4 ⋅ ( ) or {3,3} and so on. <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Pascal%27s_triangle_5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/300px-Pascal%27s_triangle_5.svg.png" decoding="async" width="300" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/450px-Pascal%27s_triangle_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/600px-Pascal%27s_triangle_5.svg.png 2x" data-file-width="540" data-file-height="389" /></a><figcaption>The numbers of faces in the above table are the same as in <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>, without the left diagonal.</figcaption></figure> </td></tr> <tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Tesseract_tetrahedron_shadow_matrices.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Tesseract_tetrahedron_shadow_matrices.svg/300px-Tesseract_tetrahedron_shadow_matrices.svg.png" decoding="async" width="300" height="395" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Tesseract_tetrahedron_shadow_matrices.svg/450px-Tesseract_tetrahedron_shadow_matrices.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Tesseract_tetrahedron_shadow_matrices.svg/600px-Tesseract_tetrahedron_shadow_matrices.svg.png 2x" data-file-width="730" data-file-height="960" /></a><figcaption>The total number of faces is always a <a href="/wiki/Power_of_two" title="Power of two">power of two</a> minus one. This figure (a projection of the <a href="/wiki/Tesseract" title="Tesseract">tesseract</a>) shows the centroids of the 15 faces of the tetrahedron.</figcaption></figure> </td></tr></tbody></table> In some conventions,<a href="#cite_note-7">[7]</a> the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as <a href="/wiki/Simplicial_homology" title="Simplicial homology">simplicial homology</a>) than to the study of polytopes. <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Symmetric_graphs_of_regular_simplices">Symmetric graphs of regular simplices</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=3" title="Edit section: Symmetric graphs of regular simplices">edit</a>]</div> These <a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygons</a> (skew orthogonal projections) show all the vertices of the regular simplex on a <a href="/wiki/Circle" title="Circle">circle</a>, and all vertex pairs connected by edges. <table class="wikitable"> <tbody><tr align="center"> <td><a href="/wiki/File:1-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/1-simplex_t0.svg/100px-1-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/1-simplex_t0.svg/150px-1-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/1-simplex_t0.svg/200px-1-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/Line_segment" title="Line segment">1</a> </td> <td><a href="/wiki/File:2-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/2-simplex_t0.svg/100px-2-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/2-simplex_t0.svg/150px-2-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/2-simplex_t0.svg/200px-2-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/Triangle" title="Triangle">2</a> </td> <td><a href="/wiki/File:3-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/3-simplex_t0.svg/100px-3-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/3-simplex_t0.svg/150px-3-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/3-simplex_t0.svg/200px-3-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/Tetrahedron" title="Tetrahedron">3</a> </td> <td><a href="/wiki/File:4-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/100px-4-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/150px-4-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/200px-4-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/5-cell" title="5-cell">4</a> </td> <td><a href="/wiki/File:5-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/5-simplex_t0.svg/100px-5-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/5-simplex_t0.svg/150px-5-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/5-simplex_t0.svg/200px-5-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/5-simplex" title="5-simplex">5</a> </td></tr> <tr align="center"> <td><a href="/wiki/File:6-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/6-simplex_t0.svg/100px-6-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/6-simplex_t0.svg/150px-6-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/6-simplex_t0.svg/200px-6-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/6-simplex" title="6-simplex">6</a> </td> <td><a href="/wiki/File:7-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/7-simplex_t0.svg/100px-7-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/7-simplex_t0.svg/150px-7-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/7-simplex_t0.svg/200px-7-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/7-simplex" title="7-simplex">7</a> </td> <td><a href="/wiki/File:8-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/8-simplex_t0.svg/100px-8-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/8-simplex_t0.svg/150px-8-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/8-simplex_t0.svg/200px-8-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/8-simplex" title="8-simplex">8</a> </td> <td><a href="/wiki/File:9-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/9-simplex_t0.svg/100px-9-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/9-simplex_t0.svg/150px-9-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/9-simplex_t0.svg/200px-9-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/9-simplex" title="9-simplex">9</a> </td> <td><a href="/wiki/File:10-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/10-simplex_t0.svg/100px-10-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/10-simplex_t0.svg/150px-10-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/10-simplex_t0.svg/200px-10-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/10-simplex" title="10-simplex">10</a> </td></tr> <tr align="center"> <td><a href="/wiki/File:11-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/11-simplex_t0.svg/100px-11-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/11-simplex_t0.svg/150px-11-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/11-simplex_t0.svg/200px-11-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/11-simplex" class="mw-redirect" title="11-simplex">11</a> </td> <td><a href="/wiki/File:12-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/12-simplex_t0.svg/100px-12-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/12-simplex_t0.svg/150px-12-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/12-simplex_t0.svg/200px-12-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/12-simplex" class="mw-redirect" title="12-simplex">12</a> </td> <td><a href="/wiki/File:13-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/13-simplex_t0.svg/100px-13-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/13-simplex_t0.svg/150px-13-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/13-simplex_t0.svg/200px-13-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/13-simplex" class="mw-redirect" title="13-simplex">13</a> </td> <td><a href="/wiki/File:14-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/14-simplex_t0.svg/100px-14-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/14-simplex_t0.svg/150px-14-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/14-simplex_t0.svg/200px-14-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/14-simplex" class="mw-redirect" title="14-simplex">14</a> </td> <td><a href="/wiki/File:15-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/15-simplex_t0.svg/100px-15-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/15-simplex_t0.svg/150px-15-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/15-simplex_t0.svg/200px-15-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/15-simplex" class="mw-redirect" title="15-simplex">15</a> </td></tr> <tr align="center"> <td><a href="/wiki/File:16-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/16-simplex_t0.svg/100px-16-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/16-simplex_t0.svg/150px-16-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/16-simplex_t0.svg/200px-16-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/16-simplex" class="mw-redirect" title="16-simplex">16</a> </td> <td><a href="/wiki/File:17-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/17-simplex_t0.svg/100px-17-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/17-simplex_t0.svg/150px-17-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/17-simplex_t0.svg/200px-17-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/17-simplex" class="mw-redirect" title="17-simplex">17</a> </td> <td><a href="/wiki/File:18-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/18-simplex_t0.svg/100px-18-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/18-simplex_t0.svg/150px-18-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/18-simplex_t0.svg/200px-18-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/18-simplex" class="mw-redirect" title="18-simplex">18</a> </td> <td><a href="/wiki/File:19-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/19-simplex_t0.svg/100px-19-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/19-simplex_t0.svg/150px-19-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/19-simplex_t0.svg/200px-19-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/19-simplex" class="mw-redirect" title="19-simplex">19</a> </td> <td><a href="/wiki/File:20-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/20-simplex_t0.svg/100px-20-simplex_t0.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/20-simplex_t0.svg/150px-20-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/20-simplex_t0.svg/200px-20-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> <a href="/wiki/20-simplex" class="mw-redirect" title="20-simplex">20</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Standard_simplex">Standard simplex</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=4" title="Edit section: Standard simplex">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:2D-simplex.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/150px-2D-simplex.svg.png" decoding="async" width="150" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/225px-2D-simplex.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/2D-simplex.svg/300px-2D-simplex.svg.png 2x" data-file-width="96" data-file-height="137" /></a><figcaption>The standard 2-simplex in R3</figcaption></figure> The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{n}=\left\{(t_{0},\dots ,t_{n})\in \mathbf {R} ^{n+1}~{\Bigg |}~\sum _{i=0}^{n}t_{i}=1{\text{ and }}t_{i}\geq 0{\text{ for }}i=0,\ldots ,n\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{n}=\left\{(t_{0},\dots ,t_{n})\in \mathbf {R} ^{n+1}~{\Bigg |}~\sum _{i=0}^{n}t_{i}=1{\text{ and }}t_{i}\geq 0{\text{ for }}i=0,\ldots ,n\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41abad6f203851276021cbc6c64aa95ddb3f9a2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.332ex; height:7.509ex;" alt="{\displaystyle \Delta ^{n}=\left\{(t_{0},\dots ,t_{n})\in \mathbf {R} ^{n+1}~{\Bigg |}~\sum _{i=0}^{n}t_{i}=1{\text{ and }}t_{i}\geq 0{\text{ for }}i=0,\ldots ,n\right\}}" />.</dd></dl> The simplex Δn lies in the <a href="/wiki/Affine_hyperplane" class="mw-redirect" title="Affine hyperplane">affine hyperplane</a> obtained by removing the restriction ti ≥ 0 in the above definition. The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where <dl><dd>e0 = (1, 0, 0, ..., 0),</dd> <dd>e1 = (0, 1, 0, ..., 0),</dd> <dd>⋮</dd> <dd>en = (0, 0, 0, ..., 1).</dd></dl> A standard simplex is an example of a <a href="/wiki/0/1-polytope" title="0/1-polytope">0/1-polytope</a>, with all coordinates as 0 or 1. It can also be seen one <a href="/wiki/Facet_(geometry)" title="Facet (geometry)">facet</a> of a regular (n + 1)-<a href="/wiki/Orthoplex" class="mw-redirect" title="Orthoplex">orthoplex</a>. There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, ..., vn) given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{0},\ldots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{0},\ldots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbedabac389e2d76c01cfd6f87db816a9e0b6e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.862ex; height:6.843ex;" alt="{\displaystyle (t_{0},\ldots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}}" /></dd></dl> The coefficients ti are called the <a href="/wiki/Barycentric_coordinates_(mathematics)" class="mw-redirect" title="Barycentric coordinates (mathematics)">barycentric coordinates</a> of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation preserving</a> or reversing. More generally, there is a canonical map from the standard <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}" />-simplex (with n vertices) onto any <a href="/wiki/Polytope" title="Polytope">polytope</a> with n vertices, given by the same equation (modifying indexing): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{1},\ldots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{1},\ldots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d4e5f476a67cdf967a8ba8c6c5779c872cfa44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.862ex; height:6.843ex;" alt="{\displaystyle (t_{1},\ldots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}}" /></dd></dl> These are known as <a href="/wiki/Generalized_barycentric_coordinates" class="mw-redirect" title="Generalized barycentric coordinates">generalized barycentric coordinates</a>, and express every polytope as the image of a simplex: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{n-1}\twoheadrightarrow P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">↠</mo> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{n-1}\twoheadrightarrow P.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16eb98da8b712483b43b95d2e1b3bc5ca134c0f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.261ex; height:2.676ex;" alt="{\displaystyle \Delta ^{n-1}\twoheadrightarrow P.}" /> A commonly used function from Rn to the interior of the standard <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}" />-simplex is the <a href="/wiki/Softmax_function" title="Softmax function">softmax function</a>, or normalized exponential function; this generalizes the <a href="/wiki/Standard_logistic_function" class="mw-redirect" title="Standard logistic function">standard logistic function</a>. <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=5" title="Edit section: Examples">edit</a>]</div> <ul><li>Δ0 is the point 1 in R1.</li> <li>Δ1 is the line segment joining (1, 0) and (0, 1) in R2.</li> <li>Δ2 is the <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in R3.</li> <li>Δ3 is the <a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedron</a> with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in R4.</li> <li>Δ4 is the regular <a href="/wiki/5-cell" title="5-cell">5-cell</a> with vertices (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0) and (0, 0, 0, 0, 1) in R5.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Increasing_coordinates">Increasing coordinates</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=6" title="Edit section: Increasing coordinates">edit</a>]</div> An alternative coordinate system is given by taking the <a href="/wiki/Indefinite_sum" title="Indefinite sum">indefinite sum</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{0}&=0\\s_{1}&=s_{0}+t_{0}=t_{0}\\s_{2}&=s_{1}+t_{1}=t_{0}+t_{1}\\s_{3}&=s_{2}+t_{2}=t_{0}+t_{1}+t_{2}\\&\;\;\vdots \\s_{n}&=s_{n-1}+t_{n-1}=t_{0}+t_{1}+\cdots +t_{n-1}\\s_{n+1}&=s_{n}+t_{n}=t_{0}+t_{1}+\cdots +t_{n}=1\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> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{0}&=0\\s_{1}&=s_{0}+t_{0}=t_{0}\\s_{2}&=s_{1}+t_{1}=t_{0}+t_{1}\\s_{3}&=s_{2}+t_{2}=t_{0}+t_{1}+t_{2}\\&\;\;\vdots \\s_{n}&=s_{n-1}+t_{n-1}=t_{0}+t_{1}+\cdots +t_{n-1}\\s_{n+1}&=s_{n}+t_{n}=t_{0}+t_{1}+\cdots +t_{n}=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7503c78dfa1362f27950e231e1381df40176d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:41.957ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}s_{0}&=0\\s_{1}&=s_{0}+t_{0}=t_{0}\\s_{2}&=s_{1}+t_{1}=t_{0}+t_{1}\\s_{3}&=s_{2}+t_{2}=t_{0}+t_{1}+t_{2}\\&\;\;\vdots \\s_{n}&=s_{n-1}+t_{n-1}=t_{0}+t_{1}+\cdots +t_{n-1}\\s_{n+1}&=s_{n}+t_{n}=t_{0}+t_{1}+\cdots +t_{n}=1\end{aligned}}}" /></dd></dl> This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{*}^{n}=\left\{(s_{1},\ldots ,s_{n})\in \mathbf {R} ^{n}\mid 0=s_{0}\leq s_{1}\leq s_{2}\leq \dots \leq s_{n}\leq s_{n+1}=1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{*}^{n}=\left\{(s_{1},\ldots ,s_{n})\in \mathbf {R} ^{n}\mid 0=s_{0}\leq s_{1}\leq s_{2}\leq \dots \leq s_{n}\leq s_{n+1}=1\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1280c78a37d31829e9068d67b4d20e35164d3707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.942ex; height:2.843ex;" alt="{\displaystyle \Delta _{*}^{n}=\left\{(s_{1},\ldots ,s_{n})\in \mathbf {R} ^{n}\mid 0=s_{0}\leq s_{1}\leq s_{2}\leq \dots \leq s_{n}\leq s_{n+1}=1\right\}.}" /></dd></dl> Geometrically, this is an n-dimensional subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" /> (maximal dimension, codimension 0) rather than of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d57d19e7bda7c1461e88320e5cd59ce1da1dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.322ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{n+1}}" /> (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faaa3539cef84853aaef47d2cb6aec158229c4c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.547ex; height:2.509ex;" alt="{\displaystyle t_{i}=0,}" /> here correspond to successive coordinates being equal, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}=s_{i+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}=s_{i+1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f578b488729567eff49cdbb2b6cdf2de519c25a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.626ex; height:2.009ex;" alt="{\displaystyle s_{i}=s_{i+1},}" /> while the <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a> corresponds to the inequalities becoming strict (increasing sequences). A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domain</a> for the <a href="/wiki/Group_action" title="Group action">action</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}" /> mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume 1/n!. Alternatively, the volume can be computed by an iterated integral, whose successive integrands are 1, x, x2/2, x3/3!, ..., xn/n!. A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums. <div class="mw-heading mw-heading3"><h3 id="Projection_onto_the_standard_simplex">Projection onto the standard simplex</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=7" title="Edit section: Projection onto the standard simplex">edit</a>]</div> Especially in numerical applications of <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> a <a href="/wiki/Graphical_projection" class="mw-redirect" title="Graphical projection">projection</a> onto the standard simplex is of interest. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{i})_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p_{i})_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeae3035ce9b2c943138f9af5aeedc60048bb28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.578ex; height:2.843ex;" alt="{\displaystyle (p_{i})_{i}}" /> with possibly negative entries, the closest point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(t_{i}\right)_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(t_{i}\right)_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367f26eb29c5bdd79743fcb6ee699b7a184aa8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.248ex; height:3.009ex;" alt="{\displaystyle \left(t_{i}\right)_{i}}" /> on the simplex has coordinates <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}=\max\{p_{i}+\Delta \,,0\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}=\max\{p_{i}+\Delta \,,0\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5380d05cf61a3acc8cff42a192252426dd03c245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.364ex; height:2.843ex;" alt="{\displaystyle t_{i}=\max\{p_{i}+\Delta \,,0\},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /> is chosen such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}\max\{p_{i}+\Delta \,,0\}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}\max\{p_{i}+\Delta \,,0\}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8de708c17eb8018e5217e9a790fef42130980dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.528ex; height:3.009ex;" alt="{\textstyle \sum _{i}\max\{p_{i}+\Delta \,,0\}=1.}" /> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /> can be easily calculated from sorting pi.<a href="#cite_note-8">[8]</a> The sorting approach takes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2320768fb54880ca4356e61f60eb02a3f9d9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle O(n\log n)}" /> complexity, which can be improved to O(n) complexity via <a href="/wiki/Selection_algorithm" title="Selection algorithm">median-finding</a> algorithms.<a href="#cite_note-9">[9]</a> Projecting onto the simplex is computationally similar to projecting onto the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361ddd720474aa41cb05453e03424fb7999d3b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{1}}" /> ball. <div class="mw-heading mw-heading3"><h3 id="Corner_of_cube">Corner of cube</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=8" title="Edit section: Corner of cube">edit</a>]</div> Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{c}^{n}=\left\{(t_{1},\ldots ,t_{n})\in \mathbf {R} ^{n}~{\Bigg |}~\sum _{i=1}^{n}t_{i}\leq 1{\text{ and }}t_{i}\geq 0{\text{ for all }}i\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>i</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{c}^{n}=\left\{(t_{1},\ldots ,t_{n})\in \mathbf {R} ^{n}~{\Bigg |}~\sum _{i=1}^{n}t_{i}\leq 1{\text{ and }}t_{i}\geq 0{\text{ for all }}i\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d4312d47125892caec7eb7538fcb4a8b81c35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.468ex; height:7.509ex;" alt="{\displaystyle \Delta _{c}^{n}=\left\{(t_{1},\ldots ,t_{n})\in \mathbf {R} ^{n}~{\Bigg |}~\sum _{i=1}^{n}t_{i}\leq 1{\text{ and }}t_{i}\geq 0{\text{ for all }}i\right\}.}" /></dd></dl> This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the <a href="/wiki/Simplex_method" class="mw-redirect" title="Simplex method">simplex method</a>, which is based at the origin, and locally models a vertex on a polytope with n facets. <div class="mw-heading mw-heading2"><h2 id="Cartesian_coordinates_for_a_regular_n-dimensional_simplex_in_Rn">Cartesian coordinates for a regular n-dimensional simplex in Rn</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=9" title="Edit section: Cartesian coordinates for a regular n-dimensional simplex in Rn">edit</a>]</div> One way to write down a regular n-simplex in Rn is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c1a0cd8279cea58b0ccb583e75a0ee93975883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /3}" />; and the fact that the angle subtended through the center of the simplex by any two vertices is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(-1/n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(-1/n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d45437ff5b923750fa3b1a003e73c3d9cd94011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.555ex; height:2.843ex;" alt="{\displaystyle \arccos(-1/n)}" />. It is also possible to directly write down a particular regular n-simplex in Rn which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis vectors</a> of Rn by e1 through en. Begin with the standard (n − 1)-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular n-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form (α/n, ..., α/n) for some <a href="/wiki/Real_number" title="Real number">real number</a> α. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular n-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> for α. Solving this equation shows that there are two choices for the additional vertex: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}\left(1\pm {\sqrt {n+1}}\right)\cdot (1,\dots ,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\left(1\pm {\sqrt {n+1}}\right)\cdot (1,\dots ,1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/659f2af0b31ca76bf49b0330d67ec9918b1f35cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.722ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{n}}\left(1\pm {\sqrt {n+1}}\right)\cdot (1,\dots ,1).}" /></dd></dl> Either of these, together with the standard basis vectors, yields a regular n-simplex. The above regular n-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2}}}\mathbf {e} _{i}-{\frac {1}{n{\sqrt {2}}}}{\bigg (}1\pm {\frac {1}{\sqrt {n+1}}}{\bigg )}\cdot (1,\dots ,1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2}}}\mathbf {e} _{i}-{\frac {1}{n{\sqrt {2}}}}{\bigg (}1\pm {\frac {1}{\sqrt {n+1}}}{\bigg )}\cdot (1,\dots ,1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c893ac1f03177c2e10eb9d480de9668c9e5b8271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.361ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {2}}}\mathbf {e} _{i}-{\frac {1}{n{\sqrt {2}}}}{\bigg (}1\pm {\frac {1}{\sqrt {n+1}}}{\bigg )}\cdot (1,\dots ,1),}" /></dd></dl> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbe58b9b83f8b6ec0da570e2249323a8930ef1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.557ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n}" />, and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {1}{\sqrt {2(n+1)}}}\cdot (1,\dots ,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {1}{\sqrt {2(n+1)}}}\cdot (1,\dots ,1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8357f218fbd7246db795af8abdc4e86c8e90e377" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.976ex; height:6.509ex;" alt="{\displaystyle \pm {\frac {1}{\sqrt {2(n+1)}}}\cdot (1,\dots ,1).}" /></dd></dl> Note that there are two sets of vertices described here. One set uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /> in each calculation. The other set uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" /> in each calculation. This simplex is inscribed in a hypersphere of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n/(2(n+1))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n/(2(n+1))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b2477aff6a9941a3efdfe72f3fcfe9e1b8f9a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.06ex; height:4.843ex;" alt="{\displaystyle {\sqrt {n/(2(n+1))}}}" />. A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+n^{-1}}}\cdot \mathbf {e} _{i}-n^{-3/2}({\sqrt {n+1}}\pm 1)\cdot (1,\dots ,1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </msqrt> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+n^{-1}}}\cdot \mathbf {e} _{i}-n^{-3/2}({\sqrt {n+1}}\pm 1)\cdot (1,\dots ,1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f64fa31fd440dc42aae33bbfb1c64ae672bc4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.754ex; height:3.509ex;" alt="{\displaystyle {\sqrt {1+n^{-1}}}\cdot \mathbf {e} _{i}-n^{-3/2}({\sqrt {n+1}}\pm 1)\cdot (1,\dots ,1),}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbe58b9b83f8b6ec0da570e2249323a8930ef1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.557ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n}" />, and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm n^{-1/2}\cdot (1,\dots ,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm n^{-1/2}\cdot (1,\dots ,1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d65379ba6c023445fa2cdbd9692296d2800682f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.818ex; height:3.343ex;" alt="{\displaystyle \pm n^{-1/2}\cdot (1,\dots ,1).}" /></dd></dl> The side length of this simplex is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {2(n+1)/n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {2(n+1)/n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d15fab1d4d0a7f6bb3d2d1ced195d530ffb5b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.25ex; height:3.343ex;" alt="{\textstyle {\sqrt {2(n+1)/n}}}" />. A highly symmetric way to construct a regular n-simplex is to use a representation of the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> Zn+1 by <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a>. This is an n × n orthogonal matrix Q such that Qn+1 = I is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>, but no lower power of Q is. Applying powers of this <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> to an appropriate vector v will produce the vertices of a regular n-simplex. To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\operatorname {diag} (Q_{1},Q_{2},\dots ,Q_{k}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\operatorname {diag} (Q_{1},Q_{2},\dots ,Q_{k}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7b6f0042966e65c9af7fcd66cba764a28e6d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.582ex; height:2.843ex;" alt="{\displaystyle Q=\operatorname {diag} (Q_{1},Q_{2},\dots ,Q_{k}),}" /></dd></dl> where each Qi is orthogonal and either 2 × 2 or 1 × 1. In order for Q to have order n + 1, all of these matrices must have order <a href="/wiki/Divisor" title="Divisor">dividing</a> n + 1. Therefore each Qi is either a 1 × 1 matrix whose only entry is 1 or, if n is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a>, −1; or it is a 2 × 2 matrix of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\cos {\frac {2\pi \omega _{i}}{n+1}}&-\sin {\frac {2\pi \omega _{i}}{n+1}}\\\sin {\frac {2\pi \omega _{i}}{n+1}}&\cos {\frac {2\pi \omega _{i}}{n+1}}\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\cos {\frac {2\pi \omega _{i}}{n+1}}&-\sin {\frac {2\pi \omega _{i}}{n+1}}\\\sin {\frac {2\pi \omega _{i}}{n+1}}&\cos {\frac {2\pi \omega _{i}}{n+1}}\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09565b9a58e15ee8aa072f04c5cbdabd85b7d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:25.218ex; height:8.509ex;" alt="{\displaystyle {\begin{pmatrix}\cos {\frac {2\pi \omega _{i}}{n+1}}&-\sin {\frac {2\pi \omega _{i}}{n+1}}\\\sin {\frac {2\pi \omega _{i}}{n+1}}&\cos {\frac {2\pi \omega _{i}}{n+1}}\end{pmatrix}},}" /></dd></dl> where each ωi is an <a href="/wiki/Integer" title="Integer">integer</a> between zero and n inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices Qi form a basis for the non-trivial irreducible real representations of Zn+1, and the vector being rotated is not stabilized by any of them. In practical terms, for n <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> this means that every matrix Qi is 2 × 2, there is an equality of sets <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega _{1},n+1-\omega _{1},\dots ,\omega _{n/2},n+1-\omega _{n/2}\}=\{1,\dots ,n\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\omega _{1},n+1-\omega _{1},\dots ,\omega _{n/2},n+1-\omega _{n/2}\}=\{1,\dots ,n\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03856cc182956e950d7034413968409ff7df32ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.469ex; height:3.176ex;" alt="{\displaystyle \{\omega _{1},n+1-\omega _{1},\dots ,\omega _{n/2},n+1-\omega _{n/2}\}=\{1,\dots ,n\},}" /></dd></dl> and, for every Qi, the entries of v upon which Qi acts are not both zero. For example, when n = 4, one possible matrix is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\cos(2\pi /5)&-\sin(2\pi /5)&0&0\\\sin(2\pi /5)&\cos(2\pi /5)&0&0\\0&0&\cos(4\pi /5)&-\sin(4\pi /5)\\0&0&\sin(4\pi /5)&\cos(4\pi /5)\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\cos(2\pi /5)&-\sin(2\pi /5)&0&0\\\sin(2\pi /5)&\cos(2\pi /5)&0&0\\0&0&\cos(4\pi /5)&-\sin(4\pi /5)\\0&0&\sin(4\pi /5)&\cos(4\pi /5)\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96fc558a1dbe614c43e5f22dbaf26de049dcb57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:55.272ex; height:13.176ex;" alt="{\displaystyle {\begin{pmatrix}\cos(2\pi /5)&-\sin(2\pi /5)&0&0\\\sin(2\pi /5)&\cos(2\pi /5)&0&0\\0&0&\cos(4\pi /5)&-\sin(4\pi /5)\\0&0&\sin(4\pi /5)&\cos(4\pi /5)\end{pmatrix}}.}" /></dd></dl> Applying this to the vector (1, 0, 1, 0) results in the simplex whose vertices are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}\cos(2\pi /5)\\\sin(2\pi /5)\\\cos(4\pi /5)\\\sin(4\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(4\pi /5)\\\sin(4\pi /5)\\\cos(8\pi /5)\\\sin(8\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(6\pi /5)\\\sin(6\pi /5)\\\cos(2\pi /5)\\\sin(2\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(8\pi /5)\\\sin(8\pi /5)\\\cos(6\pi /5)\\\sin(6\pi /5)\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}\cos(2\pi /5)\\\sin(2\pi /5)\\\cos(4\pi /5)\\\sin(4\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(4\pi /5)\\\sin(4\pi /5)\\\cos(8\pi /5)\\\sin(8\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(6\pi /5)\\\sin(6\pi /5)\\\cos(2\pi /5)\\\sin(2\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(8\pi /5)\\\sin(8\pi /5)\\\cos(6\pi /5)\\\sin(6\pi /5)\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c36f4798408a80bb8cd4253ba1e2ae026a5793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:68.996ex; height:13.176ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}\cos(2\pi /5)\\\sin(2\pi /5)\\\cos(4\pi /5)\\\sin(4\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(4\pi /5)\\\sin(4\pi /5)\\\cos(8\pi /5)\\\sin(8\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(6\pi /5)\\\sin(6\pi /5)\\\cos(2\pi /5)\\\sin(2\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(8\pi /5)\\\sin(8\pi /5)\\\cos(6\pi /5)\\\sin(6\pi /5)\end{pmatrix}},}" /></dd></dl> each of which has distance √5 from the others. When n is odd, the condition means that exactly one of the diagonal blocks is 1 × 1, equal to −1, and acts upon a non-zero entry of v; while the remaining diagonal blocks, say Q1, ..., Q(n − 1) / 2, are 2 × 2, there is an equality of sets <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\omega _{1},-\omega _{1},\dots ,\omega _{(n-1)/2},-\omega _{n-1)/2}\right\}=\left\{1,\dots ,(n-1)/2,(n+3)/2,\dots ,n\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\omega _{1},-\omega _{1},\dots ,\omega _{(n-1)/2},-\omega _{n-1)/2}\right\}=\left\{1,\dots ,(n-1)/2,(n+3)/2,\dots ,n\right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9521695b521c8b9e918de7c0d70b7f72bd822db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:72.778ex; height:3.343ex;" alt="{\displaystyle \left\{\omega _{1},-\omega _{1},\dots ,\omega _{(n-1)/2},-\omega _{n-1)/2}\right\}=\left\{1,\dots ,(n-1)/2,(n+3)/2,\dots ,n\right\},}" /></dd></dl> and each diagonal block acts upon a pair of entries of v which are not both zero. So, for example, when n = 3, the matrix can be <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\\\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\\\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df623fe23e5217a5a31ab710132cc2332a2522a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.214ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\\\end{pmatrix}}.}" /></dd></dl> For the vector (1, 0, 1/√2), the resulting simplex has vertices <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\1\\-1/\surd 2\end{pmatrix}},{\begin{pmatrix}-1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\-1\\-1/\surd 2\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">√</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <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> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">√</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">√</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">√</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\1\\-1/\surd 2\end{pmatrix}},{\begin{pmatrix}-1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\-1\\-1/\surd 2\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d563b3cc1b77c708c2485a363e8203e767f7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:48.331ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\1\\-1/\surd 2\end{pmatrix}},{\begin{pmatrix}-1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\-1\\-1/\surd 2\end{pmatrix}},}" /></dd></dl> each of which has distance 2 from the others. <div class="mw-heading mw-heading2"><h2 id="Geometric_properties">Geometric properties</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=10" title="Edit section: Geometric properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Volume">Volume</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=11" title="Edit section: Volume">edit</a>]</div> The <a href="/wiki/Volume" title="Volume">volume</a> of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\left|\det {\begin{pmatrix}v_{1}-v_{0}&&v_{2}-v_{0}&&\cdots &&v_{n}-v_{0}\end{pmatrix}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd></mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd></mtd> <mtd> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\left|\det {\begin{pmatrix}v_{1}-v_{0}&&v_{2}-v_{0}&&\cdots &&v_{n}-v_{0}\end{pmatrix}}\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad5d3c6620e216c97fcb16949922047e07d0872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:60.083ex; height:5.343ex;" alt="{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\left|\det {\begin{pmatrix}v_{1}-v_{0}&&v_{2}-v_{0}&&\cdots &&v_{n}-v_{0}\end{pmatrix}}\right|}" /></dd></dl> where each column of the n × n <a href="/wiki/Determinant" title="Determinant">determinant</a> is a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> that points from vertex v0 to another vertex vk.<a href="#cite_note-10">[10]</a> This formula is particularly useful when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}" /> is the origin. The expression <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\det \left[{\begin{pmatrix}v_{1}^{\text{T}}-v_{0}^{\text{T}}\\v_{2}^{\text{T}}-v_{0}^{\text{T}}\\\vdots \\v_{n}^{\text{T}}-v_{0}^{\text{T}}\end{pmatrix}}{\begin{pmatrix}v_{1}-v_{0}&v_{2}-v_{0}&\cdots &v_{n}-v_{0}\end{pmatrix}}\right]^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo movablelimits="true" form="prefix">det</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\det \left[{\begin{pmatrix}v_{1}^{\text{T}}-v_{0}^{\text{T}}\\v_{2}^{\text{T}}-v_{0}^{\text{T}}\\\vdots \\v_{n}^{\text{T}}-v_{0}^{\text{T}}\end{pmatrix}}{\begin{pmatrix}v_{1}-v_{0}&v_{2}-v_{0}&\cdots &v_{n}-v_{0}\end{pmatrix}}\right]^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0177006bffd4044db172049192e5f0130595bf4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.556ex; margin-bottom: -0.282ex; width:70.372ex; height:15.343ex;" alt="{\displaystyle \mathrm {Volume} ={\frac {1}{n!}}\det \left[{\begin{pmatrix}v_{1}^{\text{T}}-v_{0}^{\text{T}}\\v_{2}^{\text{T}}-v_{0}^{\text{T}}\\\vdots \\v_{n}^{\text{T}}-v_{0}^{\text{T}}\end{pmatrix}}{\begin{pmatrix}v_{1}-v_{0}&v_{2}-v_{0}&\cdots &v_{n}-v_{0}\end{pmatrix}}\right]^{1/2}}" /></dd></dl> employs a <a href="/wiki/Gram_determinant" class="mw-redirect" title="Gram determinant">Gram determinant</a> and works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions, e.g., a triangle in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ee047387e551a89e8481e1a9e974dcc5fd5acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{3}}" />. A more symmetric way to compute the volume of an n-simplex in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" /> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Volume} ={1 \over n!}\left|\det {\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Volume} ={1 \over n!}\left|\det {\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}\right|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d1fac147feb47e5c9230f881d415587adccc66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.694ex; height:6.176ex;" alt="{\displaystyle \mathrm {Volume} ={1 \over n!}\left|\det {\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}\right|.}" /></dd></dl> Another common way of computing the volume of the simplex is via the <a href="/wiki/Cayley%E2%80%93Menger_determinant" title="Cayley–Menger determinant">Cayley–Menger determinant</a>, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.<a href="#cite_note-11">[11]</a> Without the 1/n! it is the formula for the volume of an n-<a href="/wiki/Parallelepiped#Parallelotope" title="Parallelepiped">parallelotope</a>. This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{0},e_{1},\ldots ,e_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{0},e_{1},\ldots ,e_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4b11d2aea52b88492012dbad3ae19f44d89bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.643ex; height:2.843ex;" alt="{\displaystyle (v_{0},e_{1},\ldots ,e_{n})}" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" />. Given a <a href="/wiki/Permutation" title="Permutation">permutation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebfec86b3f22a18f086275390917d5aaa2d8c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}}" />, call a list of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0},\ v_{1},\ldots ,v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0},\ v_{1},\ldots ,v_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c7609e707cb11a7c5a2813e74f4ec61b3e5c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.503ex; height:2.009ex;" alt="{\displaystyle v_{0},\ v_{1},\ldots ,v_{n}}" /> a n-path if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0343175285252a806ca4381dc2805f918c2b2585" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.366ex; height:2.843ex;" alt="{\displaystyle v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}}" /></dd></dl> (so there are n! n-paths and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5615ffa6233b0d09d5bafafb58a752c1e8de95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.346ex; height:2.009ex;" alt="{\displaystyle v_{n}}" /> does not depend on the permutation). The following assertions hold: If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping.<a href="#cite_note-12">[12]</a> In particular, the volume of such a simplex is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {1}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {1}{n!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0857d2acd4e8003ffb22e8d9dbd41c947df0987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.566ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {1}{n!}}.}" /></dd></dl> If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotope is the image of the unit n-hypercube by the <a href="/wiki/Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">linear isomorphism</a> that sends the canonical basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\ldots ,e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}" />. As previously, this implies that the volume of a simplex coming from a n-path is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {\det(e_{1},\ldots ,e_{n})}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Vol</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {\det(e_{1},\ldots ,e_{n})}{n!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d66278b0a83c5a1a5cddf5a6b462eb9698bb160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.181ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {Vol} (P)}{n!}}={\frac {\det(e_{1},\ldots ,e_{n})}{n!}}.}" /></dd></dl> Conversely, given an n-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{0},\ v_{1},\ v_{2},\ldots v_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{0},\ v_{1},\ v_{2},\ldots v_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05cc286b9968fdcfb19ee33396bb5ed51da475c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.074ex; height:2.843ex;" alt="{\displaystyle (v_{0},\ v_{1},\ v_{2},\ldots v_{n})}" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" />, it can be supposed that the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4926cde6416a32af940ac077a90b67e8410d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:45.673ex; height:2.343ex;" alt="{\displaystyle e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}}" /> form a basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}" />. Considering the parallelotope constructed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\ldots ,e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}" />, one sees that the previous formula is valid for every simplex. Finally, the formula at the beginning of this section is obtained by observing that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(v_{1}-v_{0},v_{2}-v_{0},\ldots ,v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(v_{1}-v_{0},v_{2}-v_{0},\ldots ,v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccaa715e321e90cb88200af7e93c296ca70c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.065ex; height:2.843ex;" alt="{\displaystyle \det(v_{1}-v_{0},v_{2}-v_{0},\ldots ,v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).}" /></dd></dl> From this formula, it follows immediately that the volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over (n+1)!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over (n+1)!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f098396c4a81b9de4ea929acc992e04e3277a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.69ex; height:6.009ex;" alt="{\displaystyle {1 \over (n+1)!}}" /></dd></dl> The volume of a regular n-simplex with unit side length is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> <mrow> <mi>n</mi> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f201b940b9f6dd34dca356886f4798400d9c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:8.17ex; height:6.843ex;" alt="{\displaystyle {\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}}" /></dd></dl> as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1/{\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e986f4fe19dfd6b3588a44c9f82825e5d9914db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.851ex; height:3.176ex;" alt="{\displaystyle x=1/{\sqrt {2}}}" />  (where the n-simplex side length is 1), and normalizing by the length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx/{\sqrt {n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx/{\sqrt {n+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/604d81cea64d47d794b7d59f64294b9c5f7aa8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.041ex; height:3.009ex;" alt="{\displaystyle dx/{\sqrt {n+1}}}" /> of the increment, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (dx/(n+1),\ldots ,dx/(n+1))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (dx/(n+1),\ldots ,dx/(n+1))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d59c137af194b4e7a23b95c1eab5e0019502c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.817ex; height:2.843ex;" alt="{\displaystyle (dx/(n+1),\ldots ,dx/(n+1))}" />, along the normal vector. <div class="mw-heading mw-heading3"><h3 id="Dihedral_angles_of_the_regular_n-simplex">Dihedral angles of the regular n-simplex</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=12" title="Edit section: Dihedral angles of the regular n-simplex">edit</a>]</div> Any two (n − 1)-dimensional faces of a regular n-dimensional simplex are themselves regular (n − 1)-dimensional simplices, and they have the same <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> of cos−1(1/n).<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a> This can be seen by noting that the center of the standard simplex is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1491c17bd450b7a5e0fa64e850c8aabcdb7ed275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.799ex; height:4.843ex;" alt="{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}" />, and the centers of its faces are coordinate permutations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbdc20578ecb9404eeb82f3c87154b4344c70c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.149ex; height:3.343ex;" alt="{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}" />. Then, by symmetry, the vector pointing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1491c17bd450b7a5e0fa64e850c8aabcdb7ed275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.799ex; height:4.843ex;" alt="{\textstyle \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbdc20578ecb9404eeb82f3c87154b4344c70c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.149ex; height:3.343ex;" alt="{\textstyle \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)}" /> is perpendicular to the faces. So the vectors normal to the faces are permutations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-n,1,\dots ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-n,1,\dots ,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd68658c84234268d3b1a379076449c5929f2123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.549ex; height:2.843ex;" alt="{\displaystyle (-n,1,\dots ,1)}" />, from which the dihedral angles are calculated. <div class="mw-heading mw-heading3"><h3 id="Simplices_with_an_"orthogonal_corner"">Simplices with an "orthogonal corner"</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=13" title="Edit section: Simplices with an "orthogonal corner"">edit</a>]</div> An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an n-dimensional version of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: The sum of the squared (n − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n − 1)-dimensional volume of the facet opposite of the orthogonal corner. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13afbe9900f374f37906a8b70ec6a0834cd1b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.166ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}\ldots A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\ldots A_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d465b810fd1a5129df1400e69d696cf7ccacc07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.256ex; height:2.509ex;" alt="{\displaystyle A_{1}\ldots A_{n}}" /> are facets being pairwise orthogonal to each other but not orthogonal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c98dbd929292d464de6942e95db306b8b8a6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{0}}" />, which is the facet opposite the orthogonal corner.<a href="#cite_note-15">[15]</a> For a 2-simplex, the theorem is the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> for triangles with a right angle and for a 3-simplex it is <a href="/wiki/De_Gua%27s_theorem" title="De Gua's theorem">de Gua's theorem</a> for a tetrahedron with an orthogonal corner. <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_(n_+_1)-hypercube">Relation to the (n + 1)-hypercube</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=14" title="Edit section: Relation to the (n + 1)-hypercube">edit</a>]</div> The <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of the face lattice of an n-simplex is isomorphic to the graph of the (n + 1)-<a href="/wiki/Hypercube" title="Hypercube">hypercube</a>'s edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive. The n-simplex is also the <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> of the (n + 1)-hypercube. It is also the <a href="/wiki/Facet_(geometry)" title="Facet (geometry)">facet</a> of the (n + 1)-<a href="/wiki/Orthoplex" class="mw-redirect" title="Orthoplex">orthoplex</a>. <div class="mw-heading mw-heading3"><h3 id="Topology">Topology</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=15" title="Edit section: Topology">edit</a>]</div> <a href="/wiki/Topology" title="Topology">Topologically</a>, an n-simplex is <a href="/wiki/Topologically_equivalent" class="mw-redirect" title="Topologically equivalent">equivalent</a> to an <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">n-ball</a>. Every n-simplex is an n-dimensional <a href="/wiki/Manifold_with_corners" class="mw-redirect" title="Manifold with corners">manifold with corners</a>. <div class="mw-heading mw-heading3"><h3 id="Probability">Probability</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=16" title="Edit section: Probability">edit</a>]</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/Categorical_distribution" title="Categorical distribution">Categorical distribution</a></div> In probability theory, the points of the standard n-simplex in (n + 1)-space form the space of possible probability distributions on a finite set consisting of n + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (n + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose <a href="/wiki/Barycentric_coordinates" class="mw-redirect" title="Barycentric coordinates">barycentric coordinates</a> are precisely those probabilities. That is, the kth vertex of the simplex is assigned to have the kth probability of the (n + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism. <div class="mw-heading mw-heading3"><h3 id="Aitchison_geometry">Aitchison geometry</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=17" title="Edit section: Aitchison geometry">edit</a>]</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/Aitchison_geometry" class="mw-redirect" title="Aitchison geometry">Aitchison geometry</a></div> Aitchinson geometry is a natural way to construct an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> from the standard simplex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{D-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{D-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869dd309334020dd9139100fd526d15ad47729a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.629ex; height:2.676ex;" alt="{\displaystyle \Delta ^{D-1}}" />. It defines the following operations on simplices and real numbers: <dl><dt>Perturbation (addition)</dt></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\oplus y=\left[{\frac {x_{1}y_{1}}{\sum _{i=1}^{D}x_{i}y_{i}}},{\frac {x_{2}y_{2}}{\sum _{i=1}^{D}x_{i}y_{i}}},\dots ,{\frac {x_{D}y_{D}}{\sum _{i=1}^{D}x_{i}y_{i}}}\right]\qquad \forall x,y\in \Delta ^{D-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊕</mo> <mi>y</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="2em"></mspace> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\oplus y=\left[{\frac {x_{1}y_{1}}{\sum _{i=1}^{D}x_{i}y_{i}}},{\frac {x_{2}y_{2}}{\sum _{i=1}^{D}x_{i}y_{i}}},\dots ,{\frac {x_{D}y_{D}}{\sum _{i=1}^{D}x_{i}y_{i}}}\right]\qquad \forall x,y\in \Delta ^{D-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc3304f8174a2fc2e5c9660bad8398cedb2025e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.597ex; height:7.509ex;" alt="{\displaystyle x\oplus y=\left[{\frac {x_{1}y_{1}}{\sum _{i=1}^{D}x_{i}y_{i}}},{\frac {x_{2}y_{2}}{\sum _{i=1}^{D}x_{i}y_{i}}},\dots ,{\frac {x_{D}y_{D}}{\sum _{i=1}^{D}x_{i}y_{i}}}\right]\qquad \forall x,y\in \Delta ^{D-1}}" /></dd></dl></dd></dl> <dl><dt>Powering (scalar multiplication)</dt></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \odot x=\left[{\frac {x_{1}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}},{\frac {x_{2}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}},\ldots ,{\frac {x_{D}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}}\right]\qquad \forall x\in \Delta ^{D-1},\;\alpha \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>⊙</mo> <mi>x</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="2em"></mspace> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \odot x=\left[{\frac {x_{1}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}},{\frac {x_{2}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}},\ldots ,{\frac {x_{D}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}}\right]\qquad \forall x\in \Delta ^{D-1},\;\alpha \in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b305c5e6a917ed9e9b450ee7133c75217b054fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:69.062ex; height:7.676ex;" alt="{\displaystyle \alpha \odot x=\left[{\frac {x_{1}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}},{\frac {x_{2}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}},\ldots ,{\frac {x_{D}^{\alpha }}{\sum _{i=1}^{D}x_{i}^{\alpha }}}\right]\qquad \forall x\in \Delta ^{D-1},\;\alpha \in \mathbb {R} }" /></dd></dl></dd></dl> <dl><dt>Inner product</dt></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle ={\frac {1}{2D}}\sum _{i=1}^{D}\sum _{j=1}^{D}\log {\frac {x_{i}}{x_{j}}}\log {\frac {y_{i}}{y_{j}}}\qquad \forall x,y\in \Delta ^{D-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>D</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </munderover> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="2em"></mspace> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle ={\frac {1}{2D}}\sum _{i=1}^{D}\sum _{j=1}^{D}\log {\frac {x_{i}}{x_{j}}}\log {\frac {y_{i}}{y_{j}}}\qquad \forall x,y\in \Delta ^{D-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a34c83568b528f6231449f68b1f285840ededfa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.213ex; height:7.676ex;" alt="{\displaystyle \langle x,y\rangle ={\frac {1}{2D}}\sum _{i=1}^{D}\sum _{j=1}^{D}\log {\frac {x_{i}}{x_{j}}}\log {\frac {y_{i}}{y_{j}}}\qquad \forall x,y\in \Delta ^{D-1}}" /></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Compounds">Compounds</h3>[<a href="/w/index.php?title=Simplex&action=edit&section=18" title="Edit section: Compounds">edit</a>]</div> Since all simplices are self-dual, they can form a series of compounds; <ul><li>Two triangles form a <a href="/wiki/Hexagram" title="Hexagram">hexagram</a> {6/2}.</li> <li>Two tetrahedra form a <a href="/wiki/Compound_of_two_tetrahedra" title="Compound of two tetrahedra">compound of two tetrahedra</a> or <a href="/wiki/Stellated_octahedron" title="Stellated octahedron">stella octangula</a>.</li> <li>Two 5-cells form a <a href="/wiki/Compound_of_two_5-cells" class="mw-redirect" title="Compound of two 5-cells">compound of two 5-cells</a> in four dimensions.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Algebraic_topology">Algebraic topology</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=19" title="Edit section: Algebraic topology">edit</a>]</div> In <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, simplices are used as building blocks to construct an interesting class of <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> called <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complexes</a>. These spaces are built from simplices glued together in a <a href="/wiki/Combinatorics" title="Combinatorics">combinatorial</a> fashion. Simplicial complexes are used to define a certain kind of <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> called <a href="/wiki/Simplicial_homology" title="Simplicial homology">simplicial homology</a>. A finite set of k-simplexes embedded in an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of Rn is called an affine k-<a href="/wiki/Chain_(algebraic_topology)" title="Chain (algebraic topology)">chain</a>. The simplexes in a chain need not be unique; they may occur with <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a>. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite <a href="/wiki/Orientability" title="Orientability">orientation</a>, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients. Note that each facet of an n-simplex is an affine (n − 1)-simplex, and thus the <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> of an n-simplex is an affine (n − 1)-chain. Thus, if we denote one positively oriented affine simplex as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =[v_{0},v_{1},v_{2},\ldots ,v_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =[v_{0},v_{1},v_{2},\ldots ,v_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6eed53d9743b30ffa309c46571dd378240938b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.86ex; height:2.843ex;" alt="{\displaystyle \sigma =[v_{0},v_{1},v_{2},\ldots ,v_{n}]}" /></dd></dl> with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73fffa4919c0d6268f6a8d9f38c04dd3296fd0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.037ex; height:2.343ex;" alt="{\displaystyle v_{j}}" /> denoting the vertices, then the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b35e80bd80f80f484e30e56aabd45705a2a444aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.648ex; height:2.176ex;" alt="{\displaystyle \partial \sigma }" /> of σ is the chain <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>σ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2605f2e81a832f9898a4c6e0b876b7c2f9ade919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.925ex; height:7.176ex;" alt="{\displaystyle \partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}].}" /></dd></dl> It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{2}\sigma =\partial \left(\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}]\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>σ</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial ^{2}\sigma =\partial \left(\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}]\right)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c206798d0f7dcd17a3508ddc1e5273141c064153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.652ex; height:7.676ex;" alt="{\displaystyle \partial ^{2}\sigma =\partial \left(\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}]\right)=0.}" /></dd></dl> Likewise, the boundary of the boundary of a chain is zero: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{2}\rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial ^{2}\rho =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf626f00359eb0b423558b7c25da79ad8189b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.86ex; height:3.176ex;" alt="{\displaystyle \partial ^{2}\rho =0}" />. More generally, a simplex (and a chain) can be embedded into a <a href="/wiki/Manifold" title="Manifold">manifold</a> by means of smooth, differentiable map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbf {R} ^{n}\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbf {R} ^{n}\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782dab63de800824eb458681d008e486c5cf8584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.494ex; height:2.676ex;" alt="{\displaystyle f:\mathbf {R} ^{n}\to M}" />. In this case, both the summation convention for denoting the set, and the boundary operation commute with the <a href="/wiki/Embedding" title="Embedding">embedding</a>. That is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(\sum \nolimits _{i}a_{i}\sigma _{i}\right)=\sum \nolimits _{i}a_{i}f(\sigma _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msub> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(\sum \nolimits _{i}a_{i}\sigma _{i}\right)=\sum \nolimits _{i}a_{i}f(\sigma _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eecb2f90ff8bb3a2051ec5f6e98ff1d6e03d305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.024ex; height:4.843ex;" alt="{\displaystyle f\left(\sum \nolimits _{i}a_{i}\sigma _{i}\right)=\sum \nolimits _{i}a_{i}f(\sigma _{i})}" /></dd></dl> where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" /> are the integers denoting orientation and multiplicity. For the boundary operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b4e7c1cedb9564609aefd2aa2309972f455c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }" />, one has: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial f(\rho )=f(\partial \rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial f(\rho )=f(\partial \rho )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174b5ad386db5e0b55a92e5037ce6714ae13b175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.314ex; height:2.843ex;" alt="{\displaystyle \partial f(\rho )=f(\partial \rho )}" /></dd></dl> where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">map operation</a> (by definition of a map). A <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\sigma \to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>σ</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\sigma \to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d8559413f32f454f68d4680e47cb37b17d0177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.14ex; height:2.509ex;" alt="{\displaystyle f:\sigma \to X}" /> to a <a href="/wiki/Topological_space" title="Topological space">topological space</a> X is frequently referred to as a singular n-simplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)<a href="#cite_note-16">[16]</a> <div class="mw-heading mw-heading2"><h2 id="Algebraic_geometry">Algebraic geometry</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=20" title="Edit section: Algebraic geometry">edit</a>]</div> Since classical <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> allows one to talk about polynomial equations but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{n}:=\left\{x\in \mathbb {A} ^{n+1}~{\Bigg |}~\sum _{i=1}^{n+1}x_{i}=1\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{n}:=\left\{x\in \mathbb {A} ^{n+1}~{\Bigg |}~\sum _{i=1}^{n+1}x_{i}=1\right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2d0e86c3ce963f41cb29eda765ad05bf87ee0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.175ex; height:7.509ex;" alt="{\displaystyle \Delta ^{n}:=\left\{x\in \mathbb {A} ^{n+1}~{\Bigg |}~\sum _{i=1}^{n+1}x_{i}=1\right\},}" /> which equals the <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>-theoretic description <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{n}(R)=\operatorname {Spec} (R[\Delta ^{n}])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{n}(R)=\operatorname {Spec} (R[\Delta ^{n}])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fdc484c96359341ac664b837d50e4105e8ba6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.497ex; height:2.843ex;" alt="{\displaystyle \Delta _{n}(R)=\operatorname {Spec} (R[\Delta ^{n}])}" /> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[\Delta ^{n}]:=R[x_{1},\ldots ,x_{n+1}]\left/\left(1-\sum x_{i}\right)\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>:=</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mrow> <mo fence="true" stretchy="true" symmetric="true">/</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\Delta ^{n}]:=R[x_{1},\ldots ,x_{n+1}]\left/\left(1-\sum x_{i}\right)\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5bf71c450f9887a70f5769650fc6ef7a4e95a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.148ex; height:4.843ex;" alt="{\displaystyle R[\Delta ^{n}]:=R[x_{1},\ldots ,x_{n+1}]\left/\left(1-\sum x_{i}\right)\right.}" /> the ring of regular functions on the algebraic n-simplex (for any <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />). By using the same definitions as for the classical n-simplex, the n-simplices for different dimensions n assemble into one <a href="/wiki/Simplicial_object" class="mw-redirect" title="Simplicial object">simplicial object</a>, while the rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[\Delta ^{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\Delta ^{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4963c6f429983cd870b906dc479a7564aa8965fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.212ex; height:2.843ex;" alt="{\displaystyle R[\Delta ^{n}]}" /> assemble into one cosimplicial object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[\Delta ^{\bullet }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\Delta ^{\bullet }]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f88bc83050893b7e7c5e83e223a0f21494c3b1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.048ex; height:2.843ex;" alt="{\displaystyle R[\Delta ^{\bullet }]}" /> (in the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of schemes resp. rings, since the face and degeneracy maps are all polynomial). The algebraic n-simplices are used in higher <a href="/wiki/K-theory" title="K-theory">K-theory</a> and in the definition of higher <a href="/wiki/Chow_group" title="Chow group">Chow groups</a>. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=21" title="Edit section: Applications">edit</a>]</div> <ul><li>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, simplices are sample spaces of <a href="/wiki/Compositional_data" title="Compositional data">compositional data</a> and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a <a href="/wiki/Ternary_plot" title="Ternary plot">ternary plot</a>.</li> <li>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, a simplex space is often used to represent the space of probability distributions. The <a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a>, for instance, is defined on a simplex.</li> <li>In <a href="/wiki/Applied_statistics#industrial" class="mw-redirect" title="Applied statistics">industrial statistics</a>, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such <a href="/wiki/Mixture" title="Mixture">mixtures</a>, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using <a href="/wiki/Response_surface_methodology" title="Response surface methodology">response surface methodology</a>, and then a local maximum can be computed using a <a href="/wiki/Nonlinear_programming" title="Nonlinear programming">nonlinear programming</a> method, such as <a href="/wiki/Sequential_quadratic_programming" title="Sequential quadratic programming">sequential quadratic programming</a>.<a href="#cite_note-17">[17]</a></li> <li>In <a href="/wiki/Operations_research" title="Operations research">operations research</a>, <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a> problems can be solved by the <a href="/wiki/Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a> of <a href="/wiki/George_Dantzig" title="George Dantzig">George Dantzig</a>.</li> <li>In <a href="/wiki/Game_theory" title="Game theory">game theory</a>, strategies can be represented as points within a simplex. This representation simplifies the analysis of mixed strategies.</li> <li>In <a href="/wiki/Geometric_design" title="Geometric design">geometric design</a> and <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, many methods first perform simplicial <a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">triangulations</a> of the domain and then <a href="/wiki/Interpolation" title="Interpolation">fit interpolating</a> <a href="/wiki/Polynomial_and_rational_function_modeling" title="Polynomial and rational function modeling">polynomials</a> to each simplex.<a href="#cite_note-18">[18]</a></li> <li>In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, the hydrides of most elements in the <a href="/wiki/P-block" class="mw-redirect" title="P-block">p-block</a> can resemble a simplex if one is to connect each atom. <a href="/wiki/Neon" title="Neon">Neon</a> does not react with hydrogen and as such is <a href="/wiki/Monatomic_gas" title="Monatomic gas">a point</a>, <a href="/wiki/Fluorine" title="Fluorine">fluorine</a> bonds with one hydrogen atom and forms a line segment, <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> bonds with two hydrogen atoms in a <a href="/wiki/Bent_molecular_geometry" title="Bent molecular geometry">bent</a> fashion resembling a triangle, <a href="/wiki/Nitrogen" title="Nitrogen">nitrogen</a> reacts to form a <a href="/wiki/Trigonal_pyramidal_molecular_geometry" title="Trigonal pyramidal molecular geometry">tetrahedron</a>, and <a href="/wiki/Carbon" title="Carbon">carbon</a> forms <a href="/wiki/Tetrahedral_molecular_geometry" title="Tetrahedral molecular geometry">a structure</a> resembling a <a href="/wiki/Schlegel_diagram" title="Schlegel diagram">Schlegel diagram</a> of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a <a href="/wiki/Halogen" title="Halogen">halogen</a> atom.</li> <li>In some approaches to <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>, such as <a href="/wiki/Regge_calculus" title="Regge calculus">Regge calculus</a> and <a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">causal dynamical triangulations</a>, simplices are used as building blocks of discretizations of spacetime; that is, to build <a href="/wiki/Simplicial_manifold" title="Simplicial manifold">simplicial manifolds</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=22" title="Edit section: See also">edit</a>]</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" style="column-width: 12em;"> <ul><li><a href="/wiki/3-sphere" title="3-sphere">3-sphere</a></li> <li><a href="/wiki/Aitchison_geometry" class="mw-redirect" title="Aitchison geometry">Aitchison geometry</a></li> <li><a href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li> <li><a href="/wiki/Complete_graph" title="Complete graph">Complete graph</a></li> <li><a href="/wiki/Delaunay_triangulation" title="Delaunay triangulation">Delaunay triangulation</a></li> <li><a href="/wiki/Distance_geometry" title="Distance geometry">Distance geometry</a></li> <li><a href="/wiki/Geometric_primitive" title="Geometric primitive">Geometric primitive</a></li> <li><a href="/wiki/Hill_tetrahedron" title="Hill tetrahedron">Hill tetrahedron</a></li> <li><a href="/wiki/Hypersimplex" title="Hypersimplex">Hypersimplex</a></li> <li><a href="/wiki/List_of_regular_polytopes" title="List of regular polytopes">List of regular polytopes</a></li> <li><a href="/wiki/Metcalfe%27s_law" title="Metcalfe's law">Metcalfe's law</a></li> <li>Other regular n-<a href="/wiki/Polytope" title="Polytope">polytopes</a> <ul><li><a href="/wiki/Cross-polytope" title="Cross-polytope">Cross-polytope</a></li> <li><a href="/wiki/Hypercube" title="Hypercube">Hypercube</a></li> <li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li></ul></li> <li><a href="/wiki/Polytope" title="Polytope">Polytope</a></li> <li><a href="/wiki/Schl%C3%A4fli_orthoscheme" title="Schläfli orthoscheme">Schläfli orthoscheme</a></li> <li><a href="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm</a> – an optimization method with inequality constraints</li> <li><a href="/wiki/Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li> <li><a href="/wiki/Simplicial_homology" title="Simplicial homology">Simplicial homology</a></li> <li><a href="/wiki/Simplicial_set" title="Simplicial set">Simplicial set</a></li> <li><a href="/wiki/Spectrahedron" title="Spectrahedron">Spectrahedron</a></li> <li><a href="/wiki/Ternary_plot" title="Ternary plot">Ternary plot</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=23" title="Edit section: Notes">edit</a>]</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"><a href="#cite_ref-1">^</a> <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="CITEREFElte2006" class="citation book cs1"><a href="/wiki/Emanuel_Lodewijk_Elte" title="Emanuel Lodewijk Elte">Elte, E.L.</a> (2006) [1912]. "IV. five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Simon & Schuster. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4181-7968-7" title="Special:BookSources/978-1-4181-7968-7"><bdi>978-1-4181-7968-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=IV.+five+dimensional+semiregular+polytope&rft.btitle=The+Semiregular+Polytopes+of+the+Hyperspaces.&rft.pub=Simon+%26+Schuster&rft.date=2006&rft.isbn=978-1-4181-7968-7&rft.aulast=Elte&rft.aufirst=E.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-Boyd-2"><a href="#cite_ref-Boyd_2-0">^</a> <a href="#CITEREFBoydVandenberghe2004">Boyd & Vandenberghe 2004</a> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMiller" class="citation cs2">Miller, Jeff, <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/s.html">"Simplex"</a>, Earliest Known Uses of Some of the Words of Mathematics, retrieved 2018-01-08</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics&rft.atitle=Simplex&rft.aulast=Miller&rft.aufirst=Jeff&rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-FOOTNOTECoxeter1973120–124§7.2-4"><a href="#cite_ref-FOOTNOTECoxeter1973120–124§7.2_4-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 120–124, §7.2. </li> <li id="cite_note-FOOTNOTECoxeter1973120-5"><a href="#cite_ref-FOOTNOTECoxeter1973120_5-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 120. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSloane_"A135278"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A135278">"Sequence A135278 (Pascal's triangle with its left-hand edge removed)"</a>. The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA135278%26%23x20%3B%28Pascal%27s+triangle+with+its+left-hand+edge+removed%29&rft_id=https%3A%2F%2Foeis.org%2FA135278&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics) </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYunmei_ChenXiaojing_Ye2011" class="citation arxiv cs1">Yunmei Chen; Xiaojing Ye (2011). "Projection Onto A Simplex". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1101.6081">1101.6081</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.OC">math.OC</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Projection+Onto+A+Simplex&rft.date=2011&rft_id=info%3Aarxiv%2F1101.6081&rft.au=Yunmei+Chen&rft.au=Xiaojing+Ye&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMacUlanDe_Paula1989" class="citation journal cs1">MacUlan, N.; De Paula, G. G. (1989). "A linear-time median-finding algorithm for projecting a vector on the simplex of n". Operations Research Letters. 8 (4): 219. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-6377%2889%2990064-3">10.1016/0167-6377(89)90064-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Operations+Research+Letters&rft.atitle=A+linear-time+median-finding+algorithm+for+projecting+a+vector+on+the+simplex+of+n&rft.volume=8&rft.issue=4&rft.pages=219&rft.date=1989&rft_id=info%3Adoi%2F10.1016%2F0167-6377%2889%2990064-3&rft.aulast=MacUlan&rft.aufirst=N.&rft.au=De+Paula%2C+G.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> A derivation of a very similar formula can be found in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStein1966" class="citation journal cs1">Stein, P. (1966). "A Note on the Volume of a Simplex". American Mathematical Monthly. 73 (3): 299–301. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2315353">10.2307/2315353</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2315353">2315353</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+Note+on+the+Volume+of+a+Simplex&rft.volume=73&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E299-%3C%2Fspan%3E301&rft.date=1966&rft_id=info%3Adoi%2F10.2307%2F2315353&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2315353%23id-name%3DJSTOR&rft.aulast=Stein&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Colins, Karen D. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/">"Cayley-Menger Determinant"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Cayley-Menger+Determinant&rft.au=Colins%2C+Karen+D.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Every n-path corresponding to a permutation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>σ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720bb7a5cc8c5c9f7fe4873f4cd28782cba20b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.94ex; height:1.343ex;" alt="{\displaystyle \scriptstyle \sigma }" /> is the image of the n-path <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{1},\ v_{0}+e_{1}+e_{2},\ldots v_{0}+e_{1}+\cdots +e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{1},\ v_{0}+e_{1}+e_{2},\ldots v_{0}+e_{1}+\cdots +e_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9341f0f50d8100c82f9f760d67badd4c9211f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.693ex; height:1.843ex;" alt="{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{1},\ v_{0}+e_{1}+e_{2},\ldots v_{0}+e_{1}+\cdots +e_{n}}" /> by the affine isometry that sends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40cbab3a08140ad788e7046d5feb8de70a51c18b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.629ex; height:1.676ex;" alt="{\displaystyle \scriptstyle v_{0}}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40cbab3a08140ad788e7046d5feb8de70a51c18b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.629ex; height:1.676ex;" alt="{\displaystyle \scriptstyle v_{0}}" />, and whose linear part matches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle e_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda55c4a519aac6f51e885ad4b3a8b1764e895d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.391ex; height:1.509ex;" alt="{\displaystyle \scriptstyle e_{i}}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle e_{\sigma (i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle e_{\sigma (i)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae7247696a454edac9cedc94e5078bb9246f568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.193ex; height:2.009ex;" alt="{\displaystyle \scriptstyle e_{\sigma (i)}}" /> for all i. hence every two n-paths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the n-path <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{\sigma (1)},\ v_{0}+e_{\sigma (1)}+e_{\sigma (2)}\ldots v_{0}+e_{\sigma (1)}+\cdots +e_{\sigma (n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>…</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{\sigma (1)},\ v_{0}+e_{\sigma (1)}+e_{\sigma (2)}\ldots v_{0}+e_{\sigma (1)}+\cdots +e_{\sigma (n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c577fb6e1b0e16ef15a19fe5bff93aee092368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.246ex; height:2.176ex;" alt="{\displaystyle \scriptstyle v_{0},\ v_{0}+e_{\sigma (1)},\ v_{0}+e_{\sigma (1)}+e_{\sigma (2)}\ldots v_{0}+e_{\sigma (1)}+\cdots +e_{\sigma (n)}}" /> is the set of points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{0}+(x_{1}+\cdots +x_{n})e_{\sigma (1)}+\cdots +(x_{n-1}+x_{n})e_{\sigma (n-1)}+x_{n}e_{\sigma (n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{0}+(x_{1}+\cdots +x_{n})e_{\sigma (1)}+\cdots +(x_{n-1}+x_{n})e_{\sigma (n-1)}+x_{n}e_{\sigma (n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9a6df7bae5e43421a18e795c098feecaf0076b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.257ex; height:2.509ex;" alt="{\displaystyle \scriptstyle v_{0}+(x_{1}+\cdots +x_{n})e_{\sigma (1)}+\cdots +(x_{n-1}+x_{n})e_{\sigma (n-1)}+x_{n}e_{\sigma (n)}}" />, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 0<x_{i}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>0</mn> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 0<x_{i}<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e593fd8b931a5c78f67ead731cddbc046f4e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.766ex; height:1.843ex;" alt="{\displaystyle \scriptstyle 0<x_{i}<1}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle x_{1}+\cdots +x_{n}<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <mn>1.</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle x_{1}+\cdots +x_{n}<1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8326825b19fefa22be46c39e1c1f1629a49e1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.718ex; height:1.843ex;" alt="{\displaystyle \scriptstyle x_{1}+\cdots +x_{n}<1.}" /> Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are non-overlapping. The fact that the union of the simplexes is the whole unit n-hypercube follows as well, replacing the strict inequalities above by "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo>≤</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \leq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a393e5984cada103a240686f98e7a7393276f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \leq }" />". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFParksWills2002" class="citation journal cs1"><a href="/wiki/Harold_R._Parks" title="Harold R. Parks">Parks, Harold R.</a>; Wills, Dean C. (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex". 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The Mathematical Gazette. 19 (234): 206. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3605876">10.2307/3605876</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3605876">3605876</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125391795">125391795</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=1142.+An+n-dimensional+extension+of+Pythagoras%27+Theorem&rft.volume=19&rft.issue=234&rft.pages=206&rft.date=1935-07&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125391795%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3605876%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3605876&rft.aulast=Donchian&rft.aufirst=P.+S.&rft.au=Coxeter%2C+H.+S.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLee2006" class="citation book cs1">Lee, John M. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AdIRBwAAQBAJ&pg=PR1">Introduction to Topological Manifolds</a>. Springer. pp. 292–3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-22727-6" title="Special:BookSources/978-0-387-22727-6"><bdi>978-0-387-22727-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Topological+Manifolds&rft.pages=%3Cspan+class%3D%22nowrap%22%3E292-%3C%2Fspan%3E3&rft.pub=Springer&rft.date=2006&rft.isbn=978-0-387-22727-6&rft.aulast=Lee&rft.aufirst=John+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAdIRBwAAQBAJ%26pg%3DPR1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCornell,_John2002" class="citation book cs1">Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-07916-2" title="Special:BookSources/0-471-07916-2"><bdi>0-471-07916-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experiments+with+Mixtures%3A+Designs%2C+Models%2C+and+the+Analysis+of+Mixture+Data&rft.edition=third&rft.pub=Wiley&rft.date=2002&rft.isbn=0-471-07916-2&rft.au=Cornell%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVondran1998" class="citation journal cs1 cs1-prop-long-vol">Vondran, Gary L. (April 1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110607102757/http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf">"Radial and Pruned Tetrahedral Interpolation Techniques"</a> (PDF). HP Technical Report. HPL-98-95: 1–32. Archived from <a rel="nofollow" class="external text" href="http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf">the original</a> (PDF) on 2011-06-07. Retrieved 2009-11-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=HP+Technical+Report&rft.atitle=Radial+and+Pruned+Tetrahedral+Interpolation+Techniques&rft.volume=HPL-98-95&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E32&rft.date=1998-04&rft.aulast=Vondran&rft.aufirst=Gary+L.&rft_id=http%3A%2F%2Fwww.hpl.hp.com%2Ftechreports%2F98%2FHPL-98-95.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Simplex&action=edit&section=24" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudin1976" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-054235-X" title="Special:BookSources/0-07-054235-X"><bdi>0-07-054235-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=0-07-054235-X&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> (See chapter 10 for a simple review of topological properties.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTanenbaum2003" class="citation book cs1"><a href="/wiki/Andrew_S._Tanenbaum" title="Andrew S. Tanenbaum">Tanenbaum, Andrew S.</a> (2003). "§2.5.3". Computer Networks (4th ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-066102-3" title="Special:BookSources/0-13-066102-3"><bdi>0-13-066102-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A72.5.3&rft.btitle=Computer+Networks&rft.edition=4th&rft.pub=Prentice+Hall&rft.date=2003&rft.isbn=0-13-066102-3&rft.aulast=Tanenbaum&rft.aufirst=Andrew+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDevroye1986" class="citation book cs1">Devroye, Luc (1986). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090505034911/http://cg.scs.carleton.ca/~luc/rnbookindex.html">Non-Uniform Random Variate Generation</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96305-7" title="Special:BookSources/0-387-96305-7"><bdi>0-387-96305-7</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://cg.scs.carleton.ca/~luc/rnbookindex.html">the original</a> on 2009-05-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-Uniform+Random+Variate+Generation&rft.pub=Springer&rft.date=1986&rft.isbn=0-387-96305-7&rft.aulast=Devroye&rft.aufirst=Luc&rft_id=http%3A%2F%2Fcg.scs.carleton.ca%2F~luc%2Frnbookindex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"></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). <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)">Regular Polytopes</a> (3rd ed.). Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61480-8" title="Special:BookSources/0-486-61480-8"><bdi>0-486-61480-8</bdi></a>.</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.edition=3rd&rft.pub=Dover&rft.date=1973&rft.isbn=0-486-61480-8&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> <ul><li>pp. 120–121, §7.2. see illustration 7-2A</li> <li>p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Simplex.html">"Simplex"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Simplex&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSimplex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBoydVandenberghe2004" class="citation book cs1"><a href="/wiki/Stephen_P._Boyd" title="Stephen P. Boyd">Boyd, Stephen</a>; <a href="/w/index.php?title=Lieven_Vandenberghe&action=edit&redlink=1" class="new" title="Lieven Vandenberghe (page does not exist)">Vandenberghe, Lieven</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IUZdAAAAQBAJ">Convex Optimization</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-39400-1" title="Special:BookSources/978-1-107-39400-1"><bdi>978-1-107-39400-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+Optimization&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-1-107-39400-1&rft.aulast=Boyd&rft.aufirst=Stephen&rft.au=Vandenberghe%2C+Lieven&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIUZdAAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimplex" class="Z3988"> As <a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf">PDF</a></li></ul> <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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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 class="mw-selflink selflink">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 href="/wiki/Four-dimensional_space" title="Four-dimensional space">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">n-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><a href="/wiki/Category:Dimension" title="Category:Dimension">Category</a></div></td></tr></tbody></table></div> <table class="wikitable mw-collapsible"> <tbody><tr> <th colspan="13" style="background:lightsteelblue;" class="skin-invert"><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-collapse navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Polytopes" title="Template:Polytopes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polytopes" title="Template talk:Polytopes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polytopes" title="Special:EditPage/Template:Polytopes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div class="navbar-ct-mini">Fundamental convex <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a> and <a href="/wiki/Uniform_polytope" title="Uniform polytope">uniform polytopes</a> in dimensions 2–10</div> </th></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Coxeter_group#Finite_Coxeter_groups" title="Coxeter group">Family</a> </th> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group">An</a> </td> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group">Bn</a> </td> <td style="background:gainsboro;">I2(p) / <a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group">Dn</a> </td> <td style="background:gainsboro;"><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E6</a> / <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E7</a> / <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E8</a> / <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a> / <a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a> </td> <td style="background:gainsboro;"><a href="/wiki/H4_(mathematics)" class="mw-redirect" title="H4 (mathematics)">Hn</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a> </th> <td><a href="/wiki/Equilateral_triangle" title="Equilateral triangle">Triangle</a> </td> <td><a href="/wiki/Square" title="Square">Square</a> </td> <td style="background:#f0f0e0;"><a href="/wiki/Regular_polygon" title="Regular polygon">p-gon</a> </td> <td style="background:#e0e0f0;"><a href="/wiki/Hexagon" title="Hexagon">Hexagon</a> </td> <td><a href="/wiki/Pentagon" title="Pentagon">Pentagon</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">Uniform polyhedron</a> </th> <td style=""><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a> </td> <td style=""><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a> • <a href="/wiki/Cube" title="Cube">Cube</a> </td> <td style=""><a href="/wiki/Tetrahedron" title="Tetrahedron">Demicube</a> </td> <td style=""> </td> <td style=""><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Dodecahedron</a> • <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">Icosahedron</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polychoron" class="mw-redirect" title="Uniform polychoron">Uniform polychoron</a> </th> <td><a href="/wiki/5-cell" title="5-cell">Pentachoron</a> </td> <td><a href="/wiki/16-cell" title="16-cell">16-cell</a> • <a href="/wiki/Tesseract" title="Tesseract">Tesseract</a> </td> <td><a href="/wiki/16-cell" title="16-cell">Demitesseract</a> </td> <td style="background:#e0f0e0;"><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td> <td><a href="/wiki/120-cell" title="120-cell">120-cell</a> • <a href="/wiki/600-cell" title="600-cell">600-cell</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_5-polytope" title="Uniform 5-polytope">Uniform 5-polytope</a> </th> <td style=""><a href="/wiki/5-simplex" title="5-simplex">5-simplex</a> </td> <td style=""><a href="/wiki/5-orthoplex" title="5-orthoplex">5-orthoplex</a> • <a href="/wiki/5-cube" title="5-cube">5-cube</a> </td> <td style=""><a href="/wiki/5-demicube" title="5-demicube">5-demicube</a> </td> <td style=""> </td> <td style=""> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_6-polytope" title="Uniform 6-polytope">Uniform 6-polytope</a> </th> <td><a href="/wiki/6-simplex" title="6-simplex">6-simplex</a> </td> <td><a href="/wiki/6-orthoplex" title="6-orthoplex">6-orthoplex</a> • <a href="/wiki/6-cube" title="6-cube">6-cube</a> </td> <td><a href="/wiki/6-demicube" title="6-demicube">6-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_22_polytope" title="1 22 polytope">122</a> • <a href="/wiki/2_21_polytope" title="2 21 polytope">221</a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_7-polytope" title="Uniform 7-polytope">Uniform 7-polytope</a> </th> <td style=""><a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> </td> <td style=""><a href="/wiki/7-orthoplex" title="7-orthoplex">7-orthoplex</a> • <a href="/wiki/7-cube" title="7-cube">7-cube</a> </td> <td style=""><a href="/wiki/7-demicube" title="7-demicube">7-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_32_polytope" title="1 32 polytope">132</a> • <a href="/wiki/2_31_polytope" title="2 31 polytope">231</a> • <a href="/wiki/3_21_polytope" title="3 21 polytope">321</a> </td> <td style=""> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_8-polytope" title="Uniform 8-polytope">Uniform 8-polytope</a> </th> <td><a href="/wiki/8-simplex" title="8-simplex">8-simplex</a> </td> <td><a href="/wiki/8-orthoplex" title="8-orthoplex">8-orthoplex</a> • <a href="/wiki/8-cube" title="8-cube">8-cube</a> </td> <td><a href="/wiki/8-demicube" title="8-demicube">8-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_42_polytope" title="1 42 polytope">142</a> • <a href="/wiki/2_41_polytope" title="2 41 polytope">241</a> • <a href="/wiki/4_21_polytope" title="4 21 polytope">421</a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_9-polytope" title="Uniform 9-polytope">Uniform 9-polytope</a> </th> <td style=""><a href="/wiki/9-simplex" title="9-simplex">9-simplex</a> </td> <td style=""><a href="/wiki/9-orthoplex" title="9-orthoplex">9-orthoplex</a> • <a href="/wiki/9-cube" title="9-cube">9-cube</a> </td> <td style=""><a href="/wiki/9-demicube" title="9-demicube">9-demicube</a> </td> <td style=""> </td> <td style=""> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_10-polytope" title="Uniform 10-polytope">Uniform 10-polytope</a> </th> <td><a href="/wiki/10-simplex" title="10-simplex">10-simplex</a> </td> <td><a href="/wiki/10-orthoplex" title="10-orthoplex">10-orthoplex</a> • <a href="/wiki/10-cube" title="10-cube">10-cube</a> </td> <td><a href="/wiki/10-demicube" title="10-demicube">10-demicube</a> </td> <td> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;">Uniform n-<a href="/wiki/Polytope" title="Polytope">polytope</a> </th> <td style="">n-<a class="mw-selflink selflink">simplex</a> </td> <td style="">n-<a href="/wiki/Cross-polytope" title="Cross-polytope">orthoplex</a> • n-<a href="/wiki/Hypercube" title="Hypercube">cube</a> </td> <td style="">n-<a href="/wiki/Demihypercube" title="Demihypercube">demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/Uniform_1_k2_polytope" title="Uniform 1 k2 polytope">1k2</a> • <a href="/wiki/Uniform_2_k1_polytope" title="Uniform 2 k1 polytope">2k1</a> • <a href="/wiki/Uniform_k_21_polytope" title="Uniform k 21 polytope">k21</a> </td> <td style="">n-<a 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class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.migration-598fb77847-fp6q7","wgBackendResponseTime":345,"wgPageParseReport":{"limitreport":{"cputime":"0.846","walltime":"1.066","ppvisitednodes":{"value":10259,"limit":1000000},"postexpandincludesize":{"value":101814,"limit":2097152},"templateargumentsize":{"value":13212,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":82294,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 680.302 1 -total"," 24.05% 163.636 1 Template:Reflist"," 18.79% 127.803 138 Template:Math"," 14.29% 97.219 8 Template:Cite_book"," 12.98% 88.334 1 Template:Short_description"," 11.00% 74.855 1 Template:Dimension_topics"," 10.73% 72.976 1 Template:Navbox"," 8.55% 58.172 2 Template:Pagetype"," 7.56% 51.458 2 Template:Sfn"," 6.06% 41.208 1 Template:Other_uses"]},"scribunto":{"limitreport-timeusage":{"value":"0.385","limit":"10.000"},"limitreport-memusage":{"value":7209133,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBoydVandenberghe2004\"] = 1,\n [\"CITEREFCornell,_John2002\"] = 1,\n [\"CITEREFCoxeter1973\"] = 1,\n [\"CITEREFDevroye1986\"] = 1,\n [\"CITEREFDonchianCoxeter1935\"] = 1,\n [\"CITEREFElte2006\"] = 1,\n [\"CITEREFLee2006\"] = 1,\n [\"CITEREFMacUlanDe_Paula1989\"] = 1,\n [\"CITEREFMiller\"] = 1,\n [\"CITEREFParksWills2002\"] = 1,\n [\"CITEREFRudin1976\"] = 1,\n [\"CITEREFStein1966\"] = 1,\n [\"CITEREFTanenbaum2003\"] = 1,\n [\"CITEREFVondran1998\"] = 1,\n [\"CITEREFWillsParks2009\"] = 1,\n [\"CITEREFYunmei_ChenXiaojing_Ye2011\"] = 1,\n}\ntemplate_list = table#1 {\n [\"CDD\"] = 11,\n [\"Citation\"] = 1,\n [\"Cite OEIS\"] = 1,\n [\"Cite arXiv\"] = 1,\n [\"Cite book\"] = 8,\n [\"Cite journal\"] = 5,\n [\"Cite 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projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-02-19T11:40:21Z","dateModified":"2025-02-24T15:08:11Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e1\/Simplexes.jpg","headline":"generalization of the notion of a triangle or tetrahedron to arbitrary dimensions"}</script> </body> </html>

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Simplex - Wikipedia