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class="vector-toc-list"> <li id="toc-Of_an_angle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_an_angle"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Of an angle</span> </div> </a> <ul id="toc-Of_an_angle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Of_a_polytope" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_a_polytope"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Of a polytope</span> </div> </a> <ul id="toc-Of_a_polytope-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Of_a_plane_tiling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Of_a_plane_tiling"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Of a plane tiling</span> </div> </a> <ul id="toc-Of_a_plane_tiling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Principal_vertex" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Principal_vertex"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Principal vertex</span> </div> </a> <button aria-controls="toc-Principal_vertex-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Principal vertex subsection</span> </button> <ul id="toc-Principal_vertex-sublist" class="vector-toc-list"> <li id="toc-Ears" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ears"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Ears</span> </div> </a> <ul id="toc-Ears-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mouths" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mouths"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Mouths</span> </div> </a> <ul id="toc-Mouths-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Number_of_vertices_of_a_polyhedron" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Number_of_vertices_of_a_polyhedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Number of vertices of a polyhedron</span> </div> </a> <ul id="toc-Number_of_vertices_of_a_polyhedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vertices_in_computer_graphics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vertices_in_computer_graphics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vertices in computer graphics</span> </div> </a> <ul id="toc-Vertices_in_computer_graphics-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"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" 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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Vertex (geometry)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 52 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-52" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">52 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%A3%D8%B3_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9)" title="رأس (هندسة) – Arabic" lang="ar" hreflang="ar" data-title="رأس (هندسة)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/V%C3%A9rtiz" title="Vértiz – Asturian" lang="ast" hreflang="ast" data-title="Vértiz" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Esk%C3%ADna" title="Eskína – Guarani" lang="gn" hreflang="gn" data-title="Eskína" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D1%80%D1%8A%D1%85_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Връх (геометрия) – Bulgarian" lang="bg" hreflang="bg" data-title="Връх (геометрия)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/V%C3%A8rtex_(geometria)" title="Vèrtex (geometria) – Catalan" lang="ca" hreflang="ca" data-title="Vèrtex (geometria)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%C4%83%D1%80%C4%83_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8)" title="Тăрă (геометри) – Chuvash" lang="cv" hreflang="cv" data-title="Тăрă (геометри)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vrchol_(geometrie)" title="Vrchol (geometrie) – Czech" lang="cs" hreflang="cs" data-title="Vrchol (geometrie)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mukwava" title="Mukwava – Shona" lang="sn" hreflang="sn" data-title="Mukwava" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ecke" title="Ecke – German" lang="de" hreflang="de" data-title="Ecke" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tipp" title="Tipp – Estonian" lang="et" hreflang="et" data-title="Tipp" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%BF%CF%81%CF%85%CF%86%CE%AE_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Κορυφή (γεωμετρία) – Greek" lang="el" hreflang="el" data-title="Κορυφή (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/V%C3%A9rtice_(geometr%C3%ADa)" title="Vértice (geometría) – Spanish" lang="es" hreflang="es" data-title="Vértice (geometría)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vertico_(geometrio)" title="Vertico (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Vertico (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erpin_(geometria)" title="Erpin (geometria) – Basque" lang="eu" hreflang="eu" data-title="Erpin (geometria)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D8%A3%D8%B3_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="رأس (هندسه) – Persian" lang="fa" hreflang="fa" data-title="رأس (هندسه)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sommet_(g%C3%A9om%C3%A9trie)" title="Sommet (géométrie) – French" lang="fr" hreflang="fr" data-title="Sommet (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/V%C3%A9rtice" title="Vértice – Galician" lang="gl" hreflang="gl" data-title="Vértice" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%BC%AD%EC%A7%93%EC%A0%90" title="꼭짓점 – Korean" lang="ko" hreflang="ko" data-title="꼭짓점" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B6%E0%A5%80%E0%A4%B0%E0%A5%8D%E0%A4%B7_(%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF)" title="शीर्ष (ज्यामिति) – Hindi" lang="hi" hreflang="hi" data-title="शीर्ष (ज्यामिति)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vrh_(geometrija)" title="Vrh (geometrija) – Croatian" lang="hr" hreflang="hr" data-title="Vrh (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Titik_sudut" title="Titik sudut – Indonesian" lang="id" hreflang="id" data-title="Titik sudut" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Vertice_(geometria)" title="Vertice (geometria) – Italian" lang="it" hreflang="it" data-title="Vertice (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%93%D7%A7%D7%95%D7%93" title="קודקוד – Hebrew" lang="he" hreflang="he" data-title="קודקוד" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Virsotne" title="Virsotne – Latvian" lang="lv" hreflang="lv" data-title="Virsotne" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BC%D0%B5_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0)" title="Теме (геометрија) – Macedonian" lang="mk" hreflang="mk" data-title="Теме (геометрија)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bucu_(geometri)" title="Bucu (geometri) – Malay" lang="ms" hreflang="ms" data-title="Bucu (geometri)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Knooppunt_(wiskunde)" title="Knooppunt (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Knooppunt (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%A0%82%E7%82%B9" title="頂点 – Japanese" lang="ja" hreflang="ja" data-title="頂点" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hj%C3%B8rne" title="Hjørne – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hjørne" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hj%C3%B8rne_i_matematikk" title="Hjørne i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Hjørne i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Vert%C3%A8x" title="Vertèx – Occitan" lang="oc" hreflang="oc" data-title="Vertèx" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/V%C3%A9rtice" title="Vértice – Portuguese" lang="pt" hreflang="pt" data-title="Vértice" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/V%C3%A2rf_(geometrie)" title="Vârf (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Vârf (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B5%D1%80%D1%88%D0%B8%D0%BD%D0%B0_(%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Вершина (геометрия) – Russian" lang="ru" hreflang="ru" data-title="Вершина (геометрия)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Kulmi_(gjeometri)" title="Kulmi (gjeometri) – Albanian" lang="sq" hreflang="sq" data-title="Kulmi (gjeometri)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vertex" title="Vertex – Simple English" lang="en-simple" hreflang="en-simple" data-title="Vertex" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vrchol_(po%C4%8D%C3%ADta%C4%8Dov%C3%A1_grafika)" title="Vrchol (počítačová grafika) – Slovak" lang="sk" hreflang="sk" data-title="Vrchol (počítačová grafika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Prese%C4%8Di%C5%A1%C4%8De" title="Presečišče – Slovenian" lang="sl" hreflang="sl" data-title="Presečišče" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Eka" title="Eka – Silesian" lang="szl" hreflang="szl" data-title="Eka" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Dahnad" title="Dahnad – Somali" lang="so" hreflang="so" data-title="Dahnad" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%95%D8%B1_(%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95)" title="سەر (ئەندازە) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سەر (ئەندازە)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/K%C3%A4rki_(geometria)" title="Kärki (geometria) – Finnish" lang="fi" hreflang="fi" data-title="Kärki (geometria)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/H%C3%B6rn" title="Hörn – Swedish" lang="sv" hreflang="sv" data-title="Hörn" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF_(%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="உச்சி (வடிவவியல்) – Tamil" lang="ta" hreflang="ta" data-title="உச்சி (வடிவவியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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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 vertices in the geometry of curves, see <a href="/wiki/Vertex_(curve)" title="Vertex (curve)">Vertex (curve)</a>. For other uses of the word, see <a href="/wiki/Vertex_(disambiguation)" class="mw-redirect mw-disambig" title="Vertex (disambiguation)">Vertex (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Two_rays_and_one_vertex.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Two_rays_and_one_vertex.png/220px-Two_rays_and_one_vertex.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Two_rays_and_one_vertex.png/330px-Two_rays_and_one_vertex.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Two_rays_and_one_vertex.png/440px-Two_rays_and_one_vertex.png 2x" data-file-width="1049" data-file-height="792" /></a><figcaption>A vertex of an angle is the endpoint where two lines or rays come together.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>vertex</b> (<abbr title="plural">pl.</abbr>: <b>vertices</b> or <b>vertexes</b>) is a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> where two or more <a href="/wiki/Curve" title="Curve">curves</a>, <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, or <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a> meet or <a href="/wiki/Intersection_(geometry)" title="Intersection (geometry)">intersect</a>. As a consequence of this definition, the point where two lines meet to form an <a href="/wiki/Angle" title="Angle">angle</a> and the corners of <a href="/wiki/Polygon" title="Polygon">polygons</a> and <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> are vertices.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Of_an_angle">Of an angle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=2" title="Edit section: Of an angle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>vertex</i> of an <a href="/wiki/Angle" title="Angle">angle</a> is the point where two <a href="/wiki/Line_(mathematics)#Ray" class="mw-redirect" title="Line (mathematics)">rays</a> begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Euclid_4-0" class="reference"><a href="#cite_note-Euclid-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Of_a_polytope">Of a polytope</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=3" title="Edit section: Of a polytope"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vertex is a corner point of a <a href="/wiki/Polygon" title="Polygon">polygon</a>, <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a>, or other higher-dimensional <a href="/wiki/Polytope" title="Polytope">polytope</a>, formed by the <a href="/wiki/Intersection_(Euclidean_geometry)" class="mw-redirect" title="Intersection (Euclidean geometry)">intersection</a> of <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>, <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> or facets of the object.<sup id="cite_ref-Euclid_4-1" class="reference"><a href="#cite_note-Euclid-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In a polygon, a vertex is called "<a href="/wiki/Convex_set" title="Convex set">convex</a>" if the <a href="/wiki/Internal_angle" class="mw-redirect" title="Internal angle">internal angle</a> of the polygon (i.e., the <a href="/wiki/Angle" title="Angle">angle</a> formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two <a href="/wiki/Right_angle" title="Right angle">right angles</a>); otherwise, it is called "concave" or "reflex".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> More generally, a vertex of a polyhedron or polytope is convex, if the intersection of the polyhedron or polytope with a sufficiently small <a href="/wiki/Sphere" title="Sphere">sphere</a> centered at the vertex is convex, and is concave otherwise. </p><p>Polytope vertices are related to <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices of graphs</a>, in that the <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">1-skeleton</a> of a polytope is a graph, the vertices of which correspond to the vertices of the polytope,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. </p><p>However, in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the <a href="/wiki/Vertex_(curve)" title="Vertex (curve)">vertices of a curve</a>, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve, there will be a point of extreme curvature near each polygon vertex.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Of_a_plane_tiling">Of a plane tiling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=4" title="Edit section: Of a plane tiling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A vertex of a plane tiling or <a href="/wiki/Tessellation" title="Tessellation">tessellation</a> is a point where three or more tiles meet;<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological <a href="/wiki/Cell_complex" class="mw-redirect" title="Cell complex">cell complex</a>, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complexes</a> are its zero-dimensional faces. </p> <div class="mw-heading mw-heading2"><h2 id="Principal_vertex">Principal vertex</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=5" title="Edit section: Principal vertex"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Parts_of_a_simple_polygon.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Parts_of_a_simple_polygon.png/280px-Parts_of_a_simple_polygon.png" decoding="async" width="280" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Parts_of_a_simple_polygon.png/420px-Parts_of_a_simple_polygon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Parts_of_a_simple_polygon.png/560px-Parts_of_a_simple_polygon.png 2x" data-file-width="1308" data-file-height="976" /></a><figcaption>Parts of a simple polygon</figcaption></figure> <p>A polygon vertex <span class="texhtml"><i>x</i><sub>i</sub></span> of a <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygon</a> <span class="texhtml mvar" style="font-style:italic;">P</span> is a principal polygon vertex if the diagonal <span class="texhtml">[<i>x</i><sub>(i − 1)</sub>, <i>x</i><sub>(i + 1)</sub>]</span> intersects the boundary of <span class="texhtml mvar" style="font-style:italic;">P</span> only at <span class="texhtml"><i>x</i><sub>(i − 1)</sub></span> and <span class="texhtml"><i>x</i><sub>(i + 1)</sub></span>. There are two types of principal vertices: <i>ears</i> and <i>mouths</i>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ears">Ears</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=6" title="Edit section: Ears"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A principal vertex <span class="texhtml"><i>x</i><sub>i</sub></span> of a simple polygon <span class="texhtml mvar" style="font-style:italic;">P</span> is called an ear if the diagonal <span class="texhtml">[<i>x</i><sub>(i − 1)</sub>, <i>x</i><sub>(i + 1)</sub>]</span> that bridges <span class="texhtml"><i>x</i><sub>i</sub></span> lies entirely in <span class="texhtml mvar" style="font-style:italic;">P</span>. (see also <a href="/wiki/Convex_polygon" title="Convex polygon">convex polygon</a>) According to the <a href="/wiki/Two_ears_theorem" title="Two ears theorem">two ears theorem</a>, every simple polygon has at least two ears.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Mouths">Mouths</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=7" title="Edit section: Mouths"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A principal vertex <span class="texhtml"><i>x</i><sub>i</sub></span> of a simple polygon <span class="texhtml mvar" style="font-style:italic;">P</span> is called a mouth if the diagonal <span class="texhtml">[<i>x</i><sub>(i − 1)</sub>, <i>x</i><sub>(i + 1)</sub>]</span> lies outside the boundary of <span class="texhtml mvar" style="font-style:italic;">P</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Number_of_vertices_of_a_polyhedron">Number of vertices of a polyhedron</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=8" title="Edit section: Number of vertices of a polyhedron"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">convex polyhedron</a>'s surface has <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V-E+F=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V-E+F=2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8ab564115bf2f7f7d12a9f873d9c6c7a50190e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.892ex; height:2.509ex;" alt="{\displaystyle V-E+F=2,}"></span></dd></dl> <p>where <span class="texhtml"><i>V</i></span> is the number of vertices, <span class="texhtml"><i>E</i></span> is the number of <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>, and <span class="texhtml"><i>F</i></span> is the number of <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>. This equation is known as <a href="/wiki/Euler%27s_polyhedron_formula" class="mw-redirect" title="Euler's polyhedron formula">Euler's polyhedron formula</a>. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a <a href="/wiki/Cube_(geometry)" class="mw-redirect" title="Cube (geometry)">cube</a> has 12 edges and 6 faces, the formula implies that it has eight vertices. </p> <div class="mw-heading mw-heading2"><h2 id="Vertices_in_computer_graphics">Vertices in computer graphics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=9" title="Edit section: Vertices in computer graphics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vertex_(computer_graphics)" title="Vertex (computer graphics)">Vertex (computer graphics)</a></div> <p>In <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, objects are often represented as triangulated <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> in which the <a href="/wiki/Vertex_(computer_graphics)" title="Vertex (computer graphics)">object vertices</a> are associated not only with three spatial coordinates but also with other graphical information necessary to render the object correctly, such as colors, <a href="/wiki/Reflectance" title="Reflectance">reflectance</a> properties, textures, and <a href="/wiki/Vertex_normal" title="Vertex normal">surface normal</a>.<sup id="cite_ref-opengl_11-0" class="reference"><a href="#cite_note-opengl-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> These properties are used in rendering by a <a href="/wiki/Vertex_shader" class="mw-redirect" title="Vertex shader">vertex shader</a>, part of the <a href="/wiki/Vertex_pipeline" title="Vertex pipeline">vertex pipeline</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Vertex_arrangement" title="Vertex arrangement">Vertex arrangement</a></li> <li><a href="/wiki/Vertex_figure" title="Vertex figure">Vertex figure</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Vertex"><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="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/Vertex.html">"Vertex"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</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=Vertex&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FVertex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/geometry/vertices-faces-edges.html">"Vertices, Edges and Faces"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Vertices%2C+Edges+and+Faces&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fgeometry%2Fvertices-faces-edges.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://sciencing.com/vertices-math-5066316.html">"What Are Vertices in Math?"</a>. <i>Sciencing</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sciencing&rft.atitle=What+Are+Vertices+in+Math%3F&rft_id=https%3A%2F%2Fsciencing.com%2Fvertices-math-5066316.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-Euclid-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Euclid_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Euclid_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1956" class="citation book cs1"><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, Thomas L.</a> (1956). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/thirteenbooksofe00eucl"><i>The Thirteen Books of Euclid's Elements</i></a></span> (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.place=New+York&rft.edition=2nd+ed.+%5BFacsimile.+Original+publication%3A+Cambridge+University+Press%2C+1925%5D&rft.pub=Dover+Publications&rft.date=1956&rft.aulast=Heath&rft.aufirst=Thomas+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthirteenbooksofe00eucl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span> <dl><dd>(3 vols.): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60088-2" title="Special:BookSources/0-486-60088-2">0-486-60088-2</a> (vol. 1), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60089-0" title="Special:BookSources/0-486-60089-0">0-486-60089-0</a> (vol. 2), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60090-4" title="Special:BookSources/0-486-60090-4">0-486-60090-4</a> (vol. 3).</dd></dl> </span></li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJingStephansson2007" class="citation book cs1">Jing, Lanru; Stephansson, Ove (2007). <i>Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications</i>. Elsevier Science.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Discrete+Element+Methods+for+Rock+Engineering%3A+Theory+and+Applications&rft.pub=Elsevier+Science&rft.date=2007&rft.aulast=Jing&rft.aufirst=Lanru&rft.au=Stephansson%2C+Ove&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Peter_McMullen" title="Peter McMullen">Peter McMullen</a>, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81496-0" title="Special:BookSources/0-521-81496-0">0-521-81496-0</a> (Page 29)</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBobenkoSchröderSullivanZiegler2008" class="citation book cs1">Bobenko, Alexander I.; Schröder, Peter; <a href="/wiki/John_M._Sullivan_(mathematician)" title="John M. Sullivan (mathematician)">Sullivan, John M.</a>; <a href="/wiki/G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter M.</a> (2008). <i>Discrete differential geometry</i>. Birkhäuser Verlag AG. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-8620-7" title="Special:BookSources/978-3-7643-8620-7"><bdi>978-3-7643-8620-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+differential+geometry&rft.pub=Birkh%C3%A4user+Verlag+AG&rft.date=2008&rft.isbn=978-3-7643-8620-7&rft.aulast=Bobenko&rft.aufirst=Alexander+I.&rft.au=Schr%C3%B6der%2C+Peter&rft.au=Sullivan%2C+John+M.&rft.au=Ziegler%2C+G%C3%BCnter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">M.V. Jaric, ed, Introduction to the Mathematics of Quasicrystals (Aperiodicity and Order, Vol 2) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-040602-0" title="Special:BookSources/0-12-040602-0">0-12-040602-0</a>, Academic Press, 1989.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevadossO'Rourke2011" class="citation book cs1"><a href="/wiki/Satyan_Devadoss" title="Satyan Devadoss">Devadoss, Satyan</a>; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (2011). <a rel="nofollow" class="external text" href="http://cs.smith.edu/~orourke/DCG/"><i>Discrete and Computational Geometry</i></a>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14553-2" title="Special:BookSources/978-0-691-14553-2"><bdi>978-0-691-14553-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+and+Computational+Geometry&rft.pub=Princeton+University+Press&rft.date=2011&rft.isbn=978-0-691-14553-2&rft.aulast=Devadoss&rft.aufirst=Satyan&rft.au=O%27Rourke%2C+Joseph&rft_id=http%3A%2F%2Fcs.smith.edu%2F~orourke%2FDCG%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeisters1975" class="citation cs2">Meisters, G. H. (1975), "Polygons have ears", <i>The American Mathematical Monthly</i>, <b>82</b> (6): <span class="nowrap">648–</span>651, <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%2F2319703">10.2307/2319703</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/2319703">2319703</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0367792">0367792</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Polygons+have+ears&rft.volume=82&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E648-%3C%2Fspan%3E651&rft.date=1975&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0367792%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2319703%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2319703&rft.aulast=Meisters&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span>.</span> </li> <li id="cite_note-opengl-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-opengl_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristen" class="citation web cs1">Christen, Martin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190412104758/https://www.opengl.org/sdk/docs/tutorials/ClockworkCoders/attributes.php">"Clockworkcoders Tutorials: Vertex Attributes"</a>. <a href="/wiki/Khronos_Group" title="Khronos Group">Khronos Group</a>. Archived from <a rel="nofollow" class="external text" href="http://www.opengl.org/sdk/docs/tutorials/ClockworkCoders/attributes.php">the original</a> on 12 April 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">26 January</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Clockworkcoders+Tutorials%3A+Vertex+Attributes&rft.pub=Khronos+Group&rft.aulast=Christen&rft.aufirst=Martin&rft_id=http%3A%2F%2Fwww.opengl.org%2Fsdk%2Fdocs%2Ftutorials%2FClockworkCoders%2Fattributes.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_(geometry)&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Polygon_Vertex"><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/PolygonVertex.html">"Polygon Vertex"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</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=Polygon+Vertex&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPolygonVertex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Polyhedron_Vertex"><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/PolyhedronVertex.html">"Polyhedron Vertex"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</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=Polyhedron+Vertex&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPolyhedronVertex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+%28geometry%29" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Principal_Vertex"><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. 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