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<ul id="toc-Regular_icosahedra-sublist" class="vector-toc-list"> <li id="toc-Convex_regular_icosahedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convex_regular_icosahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Convex regular icosahedron</span> </div> </a> <ul id="toc-Convex_regular_icosahedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Great_icosahedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Great_icosahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Great icosahedron</span> </div> </a> <ul id="toc-Great_icosahedron-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stellated_icosahedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stellated_icosahedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Stellated icosahedra</span> </div> </a> <ul id="toc-Stellated_icosahedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pyritohedral_symmetry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pyritohedral_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Pyritohedral symmetry</span> </div> </a> <button aria-controls="toc-Pyritohedral_symmetry-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 Pyritohedral symmetry subsection</span> </button> <ul id="toc-Pyritohedral_symmetry-sublist" class="vector-toc-list"> <li id="toc-Cartesian_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Cartesian coordinates</span> </div> </a> <ul id="toc-Cartesian_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jessen's_icosahedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jessen's_icosahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Jessen's icosahedron</span> </div> </a> <ul id="toc-Jessen's_icosahedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cuboctahedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cuboctahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Cuboctahedron</span> </div> </a> <ul id="toc-Cuboctahedron-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_icosahedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_icosahedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other icosahedra</span> </div> </a> <button aria-controls="toc-Other_icosahedra-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 Other icosahedra subsection</span> </button> <ul id="toc-Other_icosahedra-sublist" class="vector-toc-list"> <li id="toc-Rhombic_icosahedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rhombic_icosahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Rhombic icosahedron</span> </div> </a> <ul id="toc-Rhombic_icosahedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pyramid_and_prism_symmetries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pyramid_and_prism_symmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Pyramid and prism symmetries</span> </div> </a> <ul id="toc-Pyramid_and_prism_symmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Johnson_solids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Johnson_solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Johnson solids</span> </div> </a> <ul id="toc-Johnson_solids-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">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> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Icosahedron</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 49 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-49" 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">49 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Ikosa%C3%ABder" title="Ikosaëder – Afrikaans" lang="af" hreflang="af" data-title="Ikosaëder" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%B4%D8%B1%D9%88%D9%86%D9%8A_%D8%B3%D8%B7%D9%88%D8%AD" title="عشروني سطوح – Arabic" lang="ar" hreflang="ar" data-title="عشروني سطوح" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0kosaedr" title="İkosaedr – Azerbaijani" lang="az" hreflang="az" data-title="İkosaedr" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BA%D0%BE%D1%81%D0%B0%D0%B5%D0%B4%D1%8A%D1%80" title="Икосаедър – Bulgarian" lang="bg" hreflang="bg" data-title="Икосаедър" 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/Icosaedre" title="Icosaedre – Catalan" lang="ca" hreflang="ca" data-title="Icosaedre" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dvacetist%C4%9Bn" title="Dvacetistěn – Czech" lang="cs" hreflang="cs" data-title="Dvacetistěn" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ikosaeder" title="Ikosaeder – Danish" lang="da" hreflang="da" data-title="Ikosaeder" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ikosaeder" title="Ikosaeder – German" lang="de" hreflang="de" data-title="Ikosaeder" 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/Ikosaeeder" title="Ikosaeeder – Estonian" lang="et" hreflang="et" data-title="Ikosaeeder" 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%95%CE%B9%CE%BA%CE%BF%CF%83%CE%AC%CE%B5%CE%B4%CF%81%CE%BF" title="Εικοσάεδρο – Greek" lang="el" hreflang="el" data-title="Εικοσάεδρο" 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/Icosaedro" title="Icosaedro – Spanish" lang="es" hreflang="es" data-title="Icosaedro" 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/Dudekedro" title="Dudekedro – Esperanto" lang="eo" hreflang="eo" data-title="Dudekedro" 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/Ikosaedro" title="Ikosaedro – Basque" lang="eu" hreflang="eu" data-title="Ikosaedro" 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%A8%DB%8C%D8%B3%D8%AA%E2%80%8C%D9%88%D8%AC%D9%87%DB%8C" title="بیست‌وجهی – Persian" lang="fa" hreflang="fa" data-title="بیست‌وجهی" 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/Icosa%C3%A8dre" title="Icosaèdre – French" lang="fr" hreflang="fr" data-title="Icosaèdre" 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/Icosaedro" title="Icosaedro – Galician" lang="gl" hreflang="gl" data-title="Icosaedro" 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/%EC%9D%B4%EC%8B%AD%EB%A9%B4%EC%B2%B4" title="이십면체 – Korean" lang="ko" hreflang="ko" data-title="이십면체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BD%D5%A1%D5%B6%D5%A1%D5%B6%D5%AB%D5%BD%D5%BF" title="Քսանանիստ – Armenian" lang="hy" hreflang="hy" data-title="Քսանանիստ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ikosaedar" title="Ikosaedar – Croatian" lang="hr" hreflang="hr" data-title="Ikosaedar" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ikosaedro" title="Ikosaedro – Ido" lang="io" hreflang="io" data-title="Ikosaedro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ikosahedron" title="Ikosahedron – Indonesian" lang="id" hreflang="id" data-title="Ikosahedron" 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/Icosaedro" title="Icosaedro – Italian" lang="it" hreflang="it" data-title="Icosaedro" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A9%D7%A8%D7%99%D7%9E%D7%95%D7%9F" title="עשרימון – Hebrew" lang="he" hreflang="he" data-title="עשרימון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BA%D0%BE%D1%81%D0%B0%D1%8D%D0%B4%D1%80" title="Икосаэдр – Kazakh" lang="kk" hreflang="kk" data-title="Икосаэдр" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Icosahedron" title="Icosahedron – Latin" lang="la" hreflang="la" data-title="Icosahedron" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ikosaedrs" title="Ikosaedrs – Latvian" lang="lv" hreflang="lv" data-title="Ikosaedrs" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ikosaedras" title="Ikosaedras – Lithuanian" lang="lt" hreflang="lt" data-title="Ikosaedras" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ikoza%C3%A9der" title="Ikozaéder – Hungarian" lang="hu" hreflang="hu" data-title="Ikozaéder" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Icosa%C3%ABder" title="Icosaëder – Dutch" lang="nl" hreflang="nl" data-title="Icosaëder" 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/%E4%BA%8C%E5%8D%81%E9%9D%A2%E4%BD%93" title="二十面体 – Japanese" lang="ja" hreflang="ja" data-title="二十面体" 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/Ikosaeder" title="Ikosaeder – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ikosaeder" 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/Ikosaeder" title="Ikosaeder – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ikosaeder" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ikosaedr" title="Ikosaedr – Uzbek" lang="uz" hreflang="uz" data-title="Ikosaedr" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Icosaedro" title="Icosaedro – Portuguese" lang="pt" hreflang="pt" data-title="Icosaedro" 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/Icosaedru" title="Icosaedru – Romanian" lang="ro" hreflang="ro" data-title="Icosaedru" 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%98%D0%BA%D0%BE%D1%81%D0%B0%D1%8D%D0%B4%D1%80" title="Икосаэдр – Russian" lang="ru" hreflang="ru" data-title="Икосаэдр" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Icosaedru" title="Icosaedru – Sicilian" lang="scn" hreflang="scn" data-title="Icosaedru" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Icosahedron" title="Icosahedron – Simple English" lang="en-simple" hreflang="en-simple" data-title="Icosahedron" 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/Dvadsa%C5%A5sten" title="Dvadsaťsten – Slovak" lang="sk" hreflang="sk" data-title="Dvadsaťsten" 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/Ikozaeder" title="Ikozaeder – Slovenian" lang="sl" hreflang="sl" data-title="Ikozaeder" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ikosaedar" title="Ikosaedar – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Ikosaedar" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ikosaedri" title="Ikosaedri – Finnish" lang="fi" hreflang="fi" data-title="Ikosaedri" 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/Ikosaeder" title="Ikosaeder – Swedish" lang="sv" hreflang="sv" data-title="Ikosaeder" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%98%D0%BA%D0%BE%D1%81%D0%B0%D1%8D%D0%B4%D1%80" title="Икосаэдр – Tatar" lang="tt" hreflang="tt" data-title="Икосаэдр" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A3%E0%B8%87%E0%B8%A2%E0%B8%B5%E0%B9%88%E0%B8%AA%E0%B8%B4%E0%B8%9A%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2" title="ทรงยี่สิบหน้า – Thai" lang="th" hreflang="th" data-title="ทรงยี่สิบหน้า" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Pseudoicosahedron&redirect=no" class="mw-redirect" title="Pseudoicosahedron">Pseudoicosahedron</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Polyhedron with 20 faces</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/220px-Icosahedron.svg.png" decoding="async" width="220" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/330px-Icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Icosahedron.svg/440px-Icosahedron.svg.png 2x" data-file-width="512" data-file-height="492" /></a><figcaption>Convex <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedral_tensegrity_structure.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Icosahedral_tensegrity_structure.png/170px-Icosahedral_tensegrity_structure.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Icosahedral_tensegrity_structure.png/255px-Icosahedral_tensegrity_structure.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Icosahedral_tensegrity_structure.png/340px-Icosahedral_tensegrity_structure.png 2x" data-file-width="2400" data-file-height="2400" /></a><figcaption>A <a href="/wiki/Tensegrity" title="Tensegrity">tensegrity</a> icosahedron</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, an <b>icosahedron</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˌ/: secondary stress follows">ˌ</span><span title="/aɪ/: 'i' in 'tide'">aɪ</span><span title="'k' in 'kind'">k</span><span title="/ɒ/: 'o' in 'body'">ɒ</span><span title="'s' in 'sigh'">s</span><span title="/ə/: 'a' in 'about'">ə</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'h' in 'hi'">h</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'d' in 'dye'">d</span><span title="'r' in 'rye'">r</span><span title="/ən/: 'on' in 'button'">ən</span></span>,<span class="wrap"> </span>-<span style="border-bottom:1px dotted"><span title="'k' in 'kind'">k</span><span title="/ə/: 'a' in 'about'">ə</span></span>-,<span class="wrap"> </span>-<span style="border-bottom:1px dotted"><span title="'k' in 'kind'">k</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>-/</a></span></span> or <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/aɪ/: 'i' in 'tide'">aɪ</span><span title="/ˌ/: secondary stress follows">ˌ</span><span title="'k' in 'kind'">k</span><span title="/ɒ/: 'o' in 'body'">ɒ</span><span title="'s' in 'sigh'">s</span><span title="/ə/: 'a' in 'about'">ə</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="'h' in 'hi'">h</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="'d' in 'dye'">d</span><span title="'r' in 'rye'">r</span><span title="/ən/: 'on' in 'button'">ən</span></span>/</a></span></span><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>) is a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> with 20 faces. The name comes from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <i> </i>εἴκοσι<i> (eíkosi)</i> 'twenty' and <i> </i>ἕδρα<i> (hédra)</i> 'seat'. The plural can be either "icosahedra" (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/-<span style="border-bottom:1px dotted"><span title="'d' in 'dye'">d</span><span title="'r' in 'rye'">r</span><span title="/ə/: 'a' in 'about'">ə</span></span>/</a></span></span>) or "icosahedrons". </p><p>There are infinitely many non-<a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> shapes of icosahedra, some of them being more symmetrical than others. The best known is the (<a href="/wiki/Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">convex</a>, non-<a href="/wiki/Stellation" title="Stellation">stellated</a>) <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a>—one of the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>—whose faces are 20 <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Regular_icosahedra">Regular icosahedra</h2></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="theader">Two kinds of regular icosahedra</div></div><div class="trow"><div class="tsingle" style="width:146px;max-width:146px"><div class="thumbimage" style="height:149px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Icosahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/144px-Icosahedron.png" decoding="async" width="144" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/216px-Icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Icosahedron.png/288px-Icosahedron.png 2x" data-file-width="911" data-file-height="952" /></a></span></div><div class="thumbcaption">Convex <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a></div></div><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage" style="height:149px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Great_icosahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/140px-Great_icosahedron.png" decoding="async" width="140" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/210px-Great_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/280px-Great_icosahedron.png 2x" data-file-width="1830" data-file-height="1954" /></a></span></div><div class="thumbcaption"><a href="/wiki/Great_icosahedron" title="Great icosahedron">Great icosahedron</a></div></div></div></div></div> <p>There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> faces with five meeting at each of its twelve vertices. Both have <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a>. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a <i>great icosahedron</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Convex_regular_icosahedron">Convex regular icosahedron</h3></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">Regular icosahedron</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Golden_rectangle_regular_icosahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Golden_rectangle_regular_icosahedron.jpg/170px-Golden_rectangle_regular_icosahedron.jpg" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Golden_rectangle_regular_icosahedron.jpg/255px-Golden_rectangle_regular_icosahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Golden_rectangle_regular_icosahedron.jpg/340px-Golden_rectangle_regular_icosahedron.jpg 2x" data-file-width="2048" data-file-height="2048" /></a><figcaption>Three interlocking <a href="/wiki/Golden_rectangle" title="Golden rectangle">golden rectangles</a> inscribed in a convex regular icosahedron</figcaption></figure> <p>The convex regular icosahedron is usually referred to simply as the <i>regular icosahedron</i>, one of the five regular <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>, and is represented by its <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex. </p><p>Its <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> is the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> {5, 3} having three regular pentagonal faces around each vertex. </p> <div class="mw-heading mw-heading3"><h3 id="Great_icosahedron">Great icosahedron</h3></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/Great_icosahedron" title="Great icosahedron">Great icosahedron</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedron-spinoza.jpg" class="mw-file-description"><img alt="A detail of Spinoza monument in Amsterdam" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Icosahedron-spinoza.jpg/170px-Icosahedron-spinoza.jpg" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Icosahedron-spinoza.jpg/255px-Icosahedron-spinoza.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Icosahedron-spinoza.jpg/340px-Icosahedron-spinoza.jpg 2x" data-file-width="3861" data-file-height="3861" /></a><figcaption>A detail of <a href="/wiki/Baruch_Spinoza" title="Baruch Spinoza">Spinoza</a> monument in <a href="/wiki/Amsterdam" title="Amsterdam">Amsterdam</a></figcaption></figure> <p>The <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a> is one of the four regular star <a href="/wiki/Kepler-Poinsot_polyhedra" class="mw-redirect" title="Kepler-Poinsot polyhedra">Kepler-Poinsot polyhedra</a>. Its <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> is {3, <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a <a href="/wiki/Pentagram" title="Pentagram">pentagram</a> rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges. </p><p>Its <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> is the <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a> {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, 3}, having three regular star pentagonal faces around each vertex. </p> <div class="mw-heading mw-heading2"><h2 id="Stellated_icosahedra">Stellated icosahedra</h2></div> <p><a href="/wiki/Stellation" title="Stellation">Stellation</a> is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. </p><p>In their book <i><a href="/wiki/The_Fifty-Nine_Icosahedra" title="The Fifty-Nine Icosahedra">The Fifty-Nine Icosahedra</a></i>, Coxeter et al. enumerated 58 such stellations of the regular icosahedron. </p><p>Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them. </p><p>Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such. </p> <table class="wikitable" width="600" style="margin: 1em auto 1em auto"> <tbody><tr> <td colspan="10" style="text-align:center;"><b>Notable <a href="/wiki/The_Fifty-Nine_Icosahedra" title="The Fifty-Nine Icosahedra">stellations of the icosahedron</a></b> </td></tr> <tr style="text-align:center;"> <td><a href="/wiki/Platonic_solid" title="Platonic solid">Regular</a> </td> <td colspan="3"><a href="/wiki/Dual_uniform_polyhedron" title="Dual uniform polyhedron">Uniform duals</a> </td> <td colspan="3"><a href="/wiki/Compound_polyhedron" class="mw-redirect" title="Compound polyhedron">Regular compounds</a> </td> <td><a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Regular star</a> </td> <td colspan="2">Others </td></tr> <tr style="text-align:center;"> <th><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">(Convex) icosahedron</a> </th> <th><a href="/wiki/Small_triambic_icosahedron" title="Small triambic icosahedron">Small triambic icosahedron</a> </th> <th><a href="/wiki/Medial_triambic_icosahedron" class="mw-redirect" title="Medial triambic icosahedron">Medial triambic icosahedron</a> </th> <th><a href="/wiki/Great_triambic_icosahedron" title="Great triambic icosahedron">Great triambic icosahedron</a> </th> <th><a href="/wiki/Compound_of_five_octahedra" title="Compound of five octahedra">Compound of five octahedra</a> </th> <th><a href="/wiki/Compound_of_five_tetrahedra" title="Compound of five tetrahedra">Compound of five tetrahedra</a> </th> <th><a href="/wiki/Compound_of_ten_tetrahedra" title="Compound of ten tetrahedra">Compound of ten tetrahedra</a> </th> <th><a href="/wiki/Great_icosahedron" title="Great icosahedron">Great icosahedron</a> </th> <th><a href="/wiki/Excavated_dodecahedron" title="Excavated dodecahedron">Excavated dodecahedron</a> </th> <th><a href="/wiki/Final_stellation_of_the_icosahedron" title="Final stellation of the icosahedron">Final stellation</a> </th></tr> <tr style="text-align:center;"> <td><span typeof="mw:File"><a href="/wiki/File:Zeroth_stellation_of_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Zeroth_stellation_of_icosahedron.svg/100px-Zeroth_stellation_of_icosahedron.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Zeroth_stellation_of_icosahedron.svg/150px-Zeroth_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Zeroth_stellation_of_icosahedron.svg/200px-Zeroth_stellation_of_icosahedron.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:First_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/First_stellation_of_icosahedron.png/100px-First_stellation_of_icosahedron.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/First_stellation_of_icosahedron.png/150px-First_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/First_stellation_of_icosahedron.png/200px-First_stellation_of_icosahedron.png 2x" data-file-width="920" data-file-height="900" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Ninth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Ninth_stellation_of_icosahedron.png/100px-Ninth_stellation_of_icosahedron.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Ninth_stellation_of_icosahedron.png/150px-Ninth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Ninth_stellation_of_icosahedron.png/200px-Ninth_stellation_of_icosahedron.png 2x" data-file-width="950" data-file-height="910" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:First_compound_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_compound_stellation_of_icosahedron.png/100px-First_compound_stellation_of_icosahedron.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_compound_stellation_of_icosahedron.png/150px-First_compound_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_compound_stellation_of_icosahedron.png/200px-First_compound_stellation_of_icosahedron.png 2x" data-file-width="900" data-file-height="900" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Second_compound_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Second_compound_stellation_of_icosahedron.png/100px-Second_compound_stellation_of_icosahedron.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Second_compound_stellation_of_icosahedron.png/150px-Second_compound_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Second_compound_stellation_of_icosahedron.png/200px-Second_compound_stellation_of_icosahedron.png 2x" data-file-width="900" data-file-height="900" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Third_compound_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Third_compound_stellation_of_icosahedron.png/100px-Third_compound_stellation_of_icosahedron.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Third_compound_stellation_of_icosahedron.png/150px-Third_compound_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Third_compound_stellation_of_icosahedron.png/200px-Third_compound_stellation_of_icosahedron.png 2x" data-file-width="899" data-file-height="899" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sixteenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/100px-Sixteenth_stellation_of_icosahedron.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/150px-Sixteenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/200px-Sixteenth_stellation_of_icosahedron.png 2x" data-file-width="940" data-file-height="900" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Third_stellation_of_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Third_stellation_of_icosahedron.svg/100px-Third_stellation_of_icosahedron.svg.png" decoding="async" width="100" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Third_stellation_of_icosahedron.svg/150px-Third_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Third_stellation_of_icosahedron.svg/200px-Third_stellation_of_icosahedron.svg.png 2x" data-file-width="560" data-file-height="600" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Seventeenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Seventeenth_stellation_of_icosahedron.png/100px-Seventeenth_stellation_of_icosahedron.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Seventeenth_stellation_of_icosahedron.png/150px-Seventeenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Seventeenth_stellation_of_icosahedron.png/200px-Seventeenth_stellation_of_icosahedron.png 2x" data-file-width="909" data-file-height="910" /></a></span> </td></tr> <tr style="text-align:center;"> <td><span typeof="mw:File"><a href="/wiki/File:Stellation_diagram_of_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stellation_diagram_of_icosahedron.svg/100px-Stellation_diagram_of_icosahedron.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stellation_diagram_of_icosahedron.svg/150px-Stellation_diagram_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stellation_diagram_of_icosahedron.svg/200px-Stellation_diagram_of_icosahedron.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Small_triambic_icosahedron_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Small_triambic_icosahedron_stellation_facets.svg/100px-Small_triambic_icosahedron_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Small_triambic_icosahedron_stellation_facets.svg/150px-Small_triambic_icosahedron_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Small_triambic_icosahedron_stellation_facets.svg/200px-Small_triambic_icosahedron_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Great_triambic_icosahedron_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Great_triambic_icosahedron_stellation_facets.svg/100px-Great_triambic_icosahedron_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Great_triambic_icosahedron_stellation_facets.svg/150px-Great_triambic_icosahedron_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Great_triambic_icosahedron_stellation_facets.svg/200px-Great_triambic_icosahedron_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Compound_of_five_octahedra_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Compound_of_five_octahedra_stellation_facets.svg/100px-Compound_of_five_octahedra_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Compound_of_five_octahedra_stellation_facets.svg/150px-Compound_of_five_octahedra_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Compound_of_five_octahedra_stellation_facets.svg/200px-Compound_of_five_octahedra_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Compound_of_five_tetrahedra_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Compound_of_five_tetrahedra_stellation_facets.svg/100px-Compound_of_five_tetrahedra_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Compound_of_five_tetrahedra_stellation_facets.svg/150px-Compound_of_five_tetrahedra_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Compound_of_five_tetrahedra_stellation_facets.svg/200px-Compound_of_five_tetrahedra_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Compound_of_ten_tetrahedra_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Compound_of_ten_tetrahedra_stellation_facets.svg/100px-Compound_of_ten_tetrahedra_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Compound_of_ten_tetrahedra_stellation_facets.svg/150px-Compound_of_ten_tetrahedra_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Compound_of_ten_tetrahedra_stellation_facets.svg/200px-Compound_of_ten_tetrahedra_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Great_icosahedron_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Great_icosahedron_stellation_facets.svg/100px-Great_icosahedron_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Great_icosahedron_stellation_facets.svg/150px-Great_icosahedron_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Great_icosahedron_stellation_facets.svg/200px-Great_icosahedron_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Excavated_dodecahedron_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Excavated_dodecahedron_stellation_facets.svg/100px-Excavated_dodecahedron_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Excavated_dodecahedron_stellation_facets.svg/150px-Excavated_dodecahedron_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Excavated_dodecahedron_stellation_facets.svg/200px-Excavated_dodecahedron_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Echidnahedron_stellation_facets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Echidnahedron_stellation_facets.svg/100px-Echidnahedron_stellation_facets.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Echidnahedron_stellation_facets.svg/150px-Echidnahedron_stellation_facets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Echidnahedron_stellation_facets.svg/200px-Echidnahedron_stellation_facets.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span> </td></tr> <tr style="text-align:center;"> <td colspan="10">The stellation process on the icosahedron creates a number of related <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> and <a href="/wiki/Compound_polyhedron" class="mw-redirect" title="Compound polyhedron">compounds</a> with <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a>. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Pyritohedral_symmetry">Pyritohedral symmetry</h2></div> <table class="wikitable" align="right" width="320"> <tbody><tr> <th bgcolor="#e7dcc3" colspan="2">Pyritohedral and tetrahedral symmetries </th></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagrams</a></td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> (pyritohedral) <span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-43-h01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/24px-Uniform_polyhedron-43-h01.svg.png" decoding="async" width="24" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/36px-Uniform_polyhedron-43-h01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/48px-Uniform_polyhedron-43-h01.svg.png 2x" data-file-width="240" data-file-height="217" /></a></span><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> (tetrahedral) <span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-33-s012.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Uniform_polyhedron-33-s012.svg/24px-Uniform_polyhedron-33-s012.svg.png" decoding="async" width="24" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Uniform_polyhedron-33-s012.svg/36px-Uniform_polyhedron-33-s012.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Uniform_polyhedron-33-s012.svg/48px-Uniform_polyhedron-33-s012.svg.png 2x" data-file-width="750" data-file-height="675" /></a></span> </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a></td> <td>s{3,4}<br />sr{3,3} or <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 s{\begin{Bmatrix}3\\3\end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad31e6c3de31a0ee70396aecb0b5ac889f09216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.491ex; height:6.176ex;" alt="{\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}"></span> </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Face_(geometry)" title="Face (geometry)">Faces</a></td> <td>20 triangles:<br />8 equilateral<br />12 isosceles </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></td> <td>30 (6 short + 24 long) </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></td> <td>12 </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/List_of_spherical_symmetry_groups#Polyhedral_symmetry" title="List of spherical symmetry groups">Symmetry group</a></td> <td><a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">T<sub>h</sub></a>, [4,3<sup>+</sup>], (3*2), order 24 </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Point_groups_in_three_dimensions#Rotation_groups" title="Point groups in three dimensions">Rotation group</a></td> <td><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">T<sub>d</sub></a>, [3,3]<sup>+</sup>, (332), order 12 </td></tr> <tr> <td bgcolor="#e7dcc3"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></td> <td><a href="/wiki/Pyritohedron" class="mw-redirect" title="Pyritohedron">Pyritohedron</a> </td></tr> <tr> <td bgcolor="#e7dcc3">Properties</td> <td><a href="/wiki/Convex_set" title="Convex set">convex</a> </td></tr> <tr> <th colspan="2"><span typeof="mw:File"><a href="/wiki/File:Pseudoicosahedron_flat.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Pseudoicosahedron_flat.png/200px-Pseudoicosahedron_flat.png" decoding="async" width="200" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Pseudoicosahedron_flat.png/300px-Pseudoicosahedron_flat.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Pseudoicosahedron_flat.png/400px-Pseudoicosahedron_flat.png 2x" data-file-width="977" data-file-height="849" /></a></span><br /><a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">Net</a> </th></tr> <tr> <td colspan="2"> <table width="320px" align="right"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Icosahedron_in_cuboctahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Icosahedron_in_cuboctahedron.png/220px-Icosahedron_in_cuboctahedron.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Icosahedron_in_cuboctahedron.png/330px-Icosahedron_in_cuboctahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Icosahedron_in_cuboctahedron.png/440px-Icosahedron_in_cuboctahedron.png 2x" data-file-width="3966" data-file-height="1974" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Icosahedron_in_cuboctahedron_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Icosahedron_in_cuboctahedron_net.png/100px-Icosahedron_in_cuboctahedron_net.png" decoding="async" width="100" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Icosahedron_in_cuboctahedron_net.png/150px-Icosahedron_in_cuboctahedron_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Icosahedron_in_cuboctahedron_net.png/200px-Icosahedron_in_cuboctahedron_net.png 2x" data-file-width="1665" data-file-height="1950" /></a></span> </td></tr> <tr> <td colspan="2">A regular icosahedron is topologically identical to a <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a> with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a <a href="/wiki/Kinematics_of_the_cuboctahedron" title="Kinematics of the cuboctahedron">kinematic transformation between cuboctahedron and icosahedron</a>. </td></tr></tbody></table> </td></tr></tbody></table> <p>A <i>regular icosahedron</i> can be distorted or marked up as a lower <a href="/wiki/Pyritohedral" class="mw-redirect" title="Pyritohedral">pyritohedral</a> symmetry,<sup id="cite_ref-Koca_2-0" class="reference"><a href="#cite_note-Koca-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Baez_3-0" class="reference"><a href="#cite_note-Baez-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and is called a <b>snub octahedron</b>, <b>snub tetratetrahedron</b>, <b>snub tetrahedron</b>, and <b>pseudo-icosahedron</b>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This can be seen as an <a href="/wiki/Alternation_(geometry)" title="Alternation (geometry)">alternated</a> <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a>. If all the triangles are <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral</a>, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently. </p><p><a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">Pyritohedral symmetry</a> has the symbol (3*2), [3<sup>+</sup>,4], with order 24. <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">Tetrahedral symmetry</a> has the symbol (332), [3,3]<sup>+</sup>, with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangles</a>. </p><p>These symmetries offer <a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagrams</a>: <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> and <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> respectively, each representing the lower symmetry to the <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span>, (*532), [5,3] <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a> of order 120. </p> <div class="mw-heading mw-heading3"><h3 id="Cartesian_coordinates">Cartesian coordinates</h3></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Truncated_octahedron_internal_rectangles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Truncated_octahedron_internal_rectangles.png/150px-Truncated_octahedron_internal_rectangles.png" decoding="async" width="150" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Truncated_octahedron_internal_rectangles.png/225px-Truncated_octahedron_internal_rectangles.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Truncated_octahedron_internal_rectangles.png/300px-Truncated_octahedron_internal_rectangles.png 2x" data-file-width="1131" data-file-height="1045" /></a><figcaption>Construction from the vertices of a <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a>, showing internal rectangles.</figcaption></figure> <p>The <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a> with <a href="/wiki/Alternation_(geometry)" title="Alternation (geometry)">alternated</a> vertices deleted. </p><p>This construction is called a <i>snub tetrahedron</i> in its regular icosahedron form, generated by the same operations carried out starting with the vector (<i>ϕ</i>, 1, 0), where <i>ϕ</i> is the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>.<sup id="cite_ref-Baez_3-1" class="reference"><a href="#cite_note-Baez-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Jessen's_icosahedron"><span id="Jessen.27s_icosahedron"></span>Jessen's icosahedron</h3></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Jessen%27s_icosahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Jessen%27s_icosahedron.svg/220px-Jessen%27s_icosahedron.svg.png" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Jessen%27s_icosahedron.svg/330px-Jessen%27s_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Jessen%27s_icosahedron.svg/440px-Jessen%27s_icosahedron.svg.png 2x" data-file-width="630" data-file-height="588" /></a><figcaption>Jessen's icosahedron</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Jessen%27s_icosahedron" title="Jessen's icosahedron">Jessen's icosahedron</a></div> <p>In Jessen's icosahedron, sometimes called <i>Jessen's orthogonal icosahedron</i>, the 12 isosceles faces are arranged differently so that the figure is non-convex and has <a href="/wiki/Right_angle" title="Right angle">right</a> <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angles</a>. </p><p>It is <a href="/wiki/Hilbert%27s_third_problem" title="Hilbert's third problem">scissors congruent</a> to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Cuboctahedron">Cuboctahedron</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:A3-P5-P3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/A3-P5-P3.gif/220px-A3-P5-P3.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/7d/A3-P5-P3.gif 1.5x" data-file-width="240" data-file-height="240" /></a><figcaption>Progressions between an <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, <a href="/wiki/Pseudoicosahedron" class="mw-redirect" title="Pseudoicosahedron">pseudoicosahedron</a>, and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kinematics_of_the_cuboctahedron" title="Kinematics of the cuboctahedron">Kinematics of the cuboctahedron</a></div> <p>A regular icosahedron is topologically identical to a <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a> with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and <a href="/wiki/Orientable_double_cover" class="mw-redirect" title="Orientable double cover">double cover</a> octahedron. Cyclical kinematic transformations among the members of this family exist. </p> <div class="mw-heading mw-heading2"><h2 id="Other_icosahedra">Other icosahedra</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rhombic_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Rhombic_icosahedron.png/220px-Rhombic_icosahedron.png" decoding="async" width="220" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Rhombic_icosahedron.png/330px-Rhombic_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Rhombic_icosahedron.png/440px-Rhombic_icosahedron.png 2x" data-file-width="900" data-file-height="799" /></a><figcaption><a href="/wiki/Rhombic_icosahedron" title="Rhombic icosahedron">Rhombic icosahedron</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Rhombic_icosahedron">Rhombic icosahedron</h3></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/Rhombic_icosahedron" title="Rhombic icosahedron">Rhombic icosahedron</a></div> <p>The <a href="/wiki/Rhombic_icosahedron" title="Rhombic icosahedron">rhombic icosahedron</a> is a <a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a> made up of 20 congruent rhombs. It can be derived from the <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a> by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not <a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Pyramid_and_prism_symmetries">Pyramid and prism symmetries</h3></div> <p>Common icosahedra with pyramid and prism symmetries include: </p> <ul><li>19-sided <a href="/wiki/Pyramid" title="Pyramid">pyramid</a> (plus 1 base = 20).</li> <li>18-sided <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prism</a> (plus 2 ends = 20).</li> <li>9-sided <a href="/wiki/Antiprism" title="Antiprism">antiprism</a> (2 sets of 9 sides + 2 ends = 20).</li> <li>10-sided <a href="/wiki/Bipyramid" title="Bipyramid">bipyramid</a> (2 sets of 10 sides = 20).</li> <li>10-sided <a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedron</a> (2 sets of 10 sides = 20).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Johnson_solids">Johnson solids</h3></div> <p>Several <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a> are icosahedra:<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> </p> <table class="wikitable" width="720"> <tbody><tr> <th>J22 </th> <th>J35 </th> <th>J36 </th> <th>J59 </th> <th>J60 </th> <th>J92 </th></tr> <tr align="center" valign="top"> <td><span typeof="mw:File"><a href="/wiki/File:Gyroelongated_triangular_cupola.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Gyroelongated_triangular_cupola.png/120px-Gyroelongated_triangular_cupola.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Gyroelongated_triangular_cupola.png/180px-Gyroelongated_triangular_cupola.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Gyroelongated_triangular_cupola.png/240px-Gyroelongated_triangular_cupola.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Gyroelongated_triangular_cupola" title="Gyroelongated triangular cupola">Gyroelongated triangular cupola</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Elongated_triangular_orthobicupola.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Elongated_triangular_orthobicupola.png/120px-Elongated_triangular_orthobicupola.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Elongated_triangular_orthobicupola.png/180px-Elongated_triangular_orthobicupola.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Elongated_triangular_orthobicupola.png/240px-Elongated_triangular_orthobicupola.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Elongated_triangular_orthobicupola" title="Elongated triangular orthobicupola">Elongated triangular orthobicupola</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Elongated_triangular_gyrobicupola.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Elongated_triangular_gyrobicupola.png/120px-Elongated_triangular_gyrobicupola.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Elongated_triangular_gyrobicupola.png/180px-Elongated_triangular_gyrobicupola.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Elongated_triangular_gyrobicupola.png/240px-Elongated_triangular_gyrobicupola.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Elongated_triangular_gyrobicupola" title="Elongated triangular gyrobicupola">Elongated triangular gyrobicupola</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Parabiaugmented_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Parabiaugmented_dodecahedron.png/120px-Parabiaugmented_dodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Parabiaugmented_dodecahedron.png/180px-Parabiaugmented_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Parabiaugmented_dodecahedron.png/240px-Parabiaugmented_dodecahedron.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Parabiaugmented_dodecahedron" title="Parabiaugmented dodecahedron">Parabiaugmented dodecahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Metabiaugmented_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Metabiaugmented_dodecahedron.png/120px-Metabiaugmented_dodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Metabiaugmented_dodecahedron.png/180px-Metabiaugmented_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Metabiaugmented_dodecahedron.png/240px-Metabiaugmented_dodecahedron.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Metabiaugmented_dodecahedron" title="Metabiaugmented dodecahedron">Metabiaugmented dodecahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Triangular_hebesphenorotunda.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Triangular_hebesphenorotunda.png/120px-Triangular_hebesphenorotunda.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Triangular_hebesphenorotunda.png/180px-Triangular_hebesphenorotunda.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Triangular_hebesphenorotunda.png/240px-Triangular_hebesphenorotunda.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Triangular_hebesphenorotunda" title="Triangular hebesphenorotunda">Triangular hebesphenorotunda</a> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Johnson_solid_22_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Johnson_solid_22_net.png/120px-Johnson_solid_22_net.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Johnson_solid_22_net.png/180px-Johnson_solid_22_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Johnson_solid_22_net.png/240px-Johnson_solid_22_net.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Johnson_solid_35_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Johnson_solid_35_net.png/120px-Johnson_solid_35_net.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Johnson_solid_35_net.png/180px-Johnson_solid_35_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Johnson_solid_35_net.png/240px-Johnson_solid_35_net.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Johnson_solid_36_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Johnson_solid_36_net.png/120px-Johnson_solid_36_net.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Johnson_solid_36_net.png/180px-Johnson_solid_36_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Johnson_solid_36_net.png/240px-Johnson_solid_36_net.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Johnson_solid_59_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Johnson_solid_59_net.png/120px-Johnson_solid_59_net.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Johnson_solid_59_net.png/180px-Johnson_solid_59_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Johnson_solid_59_net.png/240px-Johnson_solid_59_net.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Johnson_solid_60_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Johnson_solid_60_net.png/120px-Johnson_solid_60_net.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Johnson_solid_60_net.png/180px-Johnson_solid_60_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Johnson_solid_60_net.png/240px-Johnson_solid_60_net.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Johnson_solid_92_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Johnson_solid_92_net.png/120px-Johnson_solid_92_net.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Johnson_solid_92_net.png/180px-Johnson_solid_92_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Johnson_solid_92_net.png/240px-Johnson_solid_92_net.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td></tr> <tr align="center" valign="top"> <td>16 triangles<br />3 squares<br /> <br />1 hexagon </td> <td>8 triangles<br />12 squares </td> <td>8 triangles<br />12 squares </td> <td>10 triangles<br /> <br />10 pentagons </td> <td>10 triangles<br /> <br />10 pentagons </td> <td>13 triangles<br />3 squares<br />3 pentagons<br />1 hexagon </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">Truncated icosahedron</a></li> <li><a href="/wiki/600-cell" title="600-cell">600-cell</a></li> <li><a href="/wiki/Icosoku" title="Icosoku">Icosoku</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output 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.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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration 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.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="CITEREFJones2003" class="citation cs2"><a href="/wiki/Daniel_Jones_(phonetician)" title="Daniel Jones (phonetician)">Jones, Daniel</a> (2003) [1917], Peter Roach; James Hartmann; Jane Setter (eds.), <i>English Pronouncing Dictionary</i>, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-12-539683-2" title="Special:BookSources/3-12-539683-2"><bdi>3-12-539683-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=English+Pronouncing+Dictionary&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=3-12-539683-2&rft.aulast=Jones&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedron" class="Z3988"></span></span> </li> <li id="cite_note-Koca-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Koca_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKocaAl-MukhainiKocaAl_Qanobi2016" class="citation journal cs1">Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01). <a rel="nofollow" class="external text" href="https://doi.org/10.24200%2Fsqujs.vol21iss2pp139-149">"Symmetry of the Pyritohedron and Lattices"</a>. <i>Sultan Qaboos University Journal for Science [SQUJS]</i>. <b>21</b> (2): 139. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24200%2Fsqujs.vol21iss2pp139-149">10.24200/squjs.vol21iss2pp139-149</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sultan+Qaboos+University+Journal+for+Science+%5BSQUJS%5D&rft.atitle=Symmetry+of+the+Pyritohedron+and+Lattices&rft.volume=21&rft.issue=2&rft.pages=139&rft.date=2016-12-01&rft_id=info%3Adoi%2F10.24200%2Fsqujs.vol21iss2pp139-149&rft.aulast=Koca&rft.aufirst=Nazife&rft.au=Al-Mukhaini%2C+Aida&rft.au=Koca%2C+Mehmet&rft.au=Al+Qanobi%2C+Amal&rft_id=https%3A%2F%2Fdoi.org%2F10.24200%252Fsqujs.vol21iss2pp139-149&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedron" class="Z3988"></span></span> </li> <li id="cite_note-Baez-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Baez_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Baez_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 id="CITEREFJohn_Baez2011" class="citation web cs1">John Baez (September 11, 2011). <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/golden.html">"Fool's Gold"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fool%27s+Gold&rft.date=2011-09-11&rft.au=John+Baez&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fgolden.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedron" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKappraff1991" class="citation book cs1">Kappraff, Jay (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475"><i>Connections: The Geometric Bridge Between Art and Science</i></a> (2nd ed.). World Scientific. p. 475.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Connections%3A+The+Geometric+Bridge+Between+Art+and+Science&rft.pages=475&rft.edition=2nd&rft.pub=World+Scientific&rft.date=1991&rft.aulast=Kappraff&rft.aufirst=Jay&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dtz76s0ZGFiQC%26pg%3DPA475&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIcosahedron" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Icosahedron.html">Icosahedron</a> on Mathworld.</span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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href="/wiki/Template:Polyhedra" title="Template:Polyhedra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polyhedra" title="Template talk:Polyhedra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polyhedra" title="Special:EditPage/Template:Polyhedra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Polyhedra" style="font-size:114%;margin:0 4em"><a href="/wiki/Polyhedron" title="Polyhedron">Polyhedra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Listed by number of faces and type</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">1–10 faces</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/Monohedron" class="mw-redirect" title="Monohedron">Monohedron</a></li> <li><a href="/wiki/Dihedron" title="Dihedron">Dihedron</a></li> <li><a href="/wiki/Hosohedron" title="Hosohedron">Trihedron</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li> <li><a href="/wiki/Pentahedron" title="Pentahedron">Pentahedron</a></li> <li><a href="/wiki/Hexahedron" title="Hexahedron">Hexahedron</a></li> <li><a href="/wiki/Heptahedron" title="Heptahedron">Heptahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Enneahedron" title="Enneahedron">Enneahedron</a></li> <li><a href="/wiki/Decahedron" title="Decahedron">Decahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">11–20 faces</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/Hendecahedron" title="Hendecahedron">Hendecahedron</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Tridecahedron" title="Tridecahedron">Tridecahedron</a></li> <li><a href="/wiki/Tetradecahedron" title="Tetradecahedron">Tetradecahedron</a></li> <li><a href="/wiki/Pentadecahedron" title="Pentadecahedron">Pentadecahedron</a></li> <li><a href="/wiki/Hexadecahedron" title="Hexadecahedron">Hexadecahedron</a></li> <li><a href="/wiki/Heptadecahedron" title="Heptadecahedron">Heptadecahedron</a></li> <li><a href="/wiki/Octadecahedron" title="Octadecahedron">Octadecahedron</a></li> <li><a href="/wiki/Enneadecahedron" title="Enneadecahedron">Enneadecahedron</a></li> <li><a class="mw-selflink selflink">Icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">>20 faces</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/Icositetrahedron" title="Icositetrahedron">Icositetrahedron</a> (24)</li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">Triacontahedron</a> (30)</li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a> (32)</li> <li><a href="/wiki/Hexoctahedron" class="mw-redirect" title="Hexoctahedron">Hexoctahedron</a> (48)</li> <li><a href="/wiki/Hexecontahedron" title="Hexecontahedron">Hexecontahedron</a> (60)</li> <li><a href="/wiki/Rhombic_enneacontahedron" title="Rhombic enneacontahedron">Enneacontahedron</a> (90)</li> <li><a href="/wiki/Rhombic_hectotriadiohedron" title="Rhombic hectotriadiohedron">Hectotriadiohedron</a> (132)</li> <li><a href="/wiki/Skew_apeirohedron" title="Skew apeirohedron">Apeirohedron</a> (∞)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">elemental things</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/Face_(geometry)" title="Face (geometry)">face</a></li> <li><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edge</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a></li> <li><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform polyhedron</a> (two infinite groups and 75) <ul><li><a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular polyhedron</a> (9)</li> <li><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">quasiregular polyhedron</a> (16)</li> <li><a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular polyhedron</a> (two infinite groups and 50)</li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">convex polyhedron</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/Platonic_solid" title="Platonic solid">Platonic solid</a> (5)</li> <li><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solid</a> (13)</li> <li><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solid</a> (13)</li> <li><a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solid</a> (92)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">non-convex polyhedron</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/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedron</a> (4)</li> <li><a href="/wiki/Star_polyhedron" title="Star polyhedron">Star polyhedron</a> (infinite)</li> <li><a href="/wiki/Uniform_star_polyhedron" title="Uniform star polyhedron">Uniform star polyhedron</a> (57)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a>‌s</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/Prism_(geometry)" title="Prism (geometry)">prism</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprism</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Wedge_(geometry)" title="Wedge (geometry)">wedge</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a></li> <li><a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Convex_polyhedra" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Convex_polyhedron_navigator" title="Template:Convex polyhedron navigator"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Convex_polyhedron_navigator" title="Template talk:Convex polyhedron navigator"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Convex_polyhedron_navigator" title="Special:EditPage/Template:Convex polyhedron navigator"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Convex_polyhedra" style="font-size:114%;margin:0 4em">Convex <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> <span class="nobold">(<a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular</a>)</span></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/Tetrahedron#Regular_tetrahedron" title="Tetrahedron">tetrahedron</a></li> <li><a href="/wiki/Cube" title="Cube">cube</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">octahedron</a></li> <li><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a></li> <li><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a><br /><span class="nobold">(<a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></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/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li> <li><a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li> <li><a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a></li> <li><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li> <li><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li> <li><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li> <li><a href="/wiki/Snub_cube" title="Snub cube">snub cube</a></li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li> <li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li> <li><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li> <li><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a><br /><span class="nobold">(duals of Archimedean)</span></div></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/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li> <li><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li> <li><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li> <li><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li> <li><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li> <li><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li> <li><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li> <li><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</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><i><a href="/wiki/Dihedron" title="Dihedron">dihedron</a></i></li> <li><i><a href="/wiki/Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a></li> <li><a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral others</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/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></li> <li><a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedra</a></li> <li><a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramid</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Bicupola_(geometry)" class="mw-redirect" title="Bicupola (geometry)">bicupola</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Bifrustum" title="Bifrustum">bifrustum</a></li> <li><a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotunda</a></li> <li><a href="/wiki/Birotunda" title="Birotunda">birotunda</a></li> <li><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a></li> <li><a href="/wiki/Scutoid" title="Scutoid">scutoid</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Degenerate polyhedra are in <i>italics</i>.</div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority 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