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id="toc-Icosahedral_graph" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Icosahedral_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Icosahedral graph</span> </div> </a> <ul id="toc-Icosahedral_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_polyhedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related polyhedra</span> </div> </a> <button aria-controls="toc-Related_polyhedra-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 Related polyhedra subsection</span> </button> <ul id="toc-Related_polyhedra-sublist" class="vector-toc-list"> <li id="toc-In_other_Platonic_solids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_other_Platonic_solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>In other Platonic solids</span> </div> </a> <ul id="toc-In_other_Platonic_solids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stellation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stellation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Stellation</span> </div> </a> <ul id="toc-Stellation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Facetings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Facetings"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Facetings</span> </div> </a> <ul id="toc-Facetings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diminishment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diminishment"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Diminishment</span> </div> </a> <ul id="toc-Diminishment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relations_to_the_600-cell_and_other_4-polytopes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relations_to_the_600-cell_and_other_4-polytopes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Relations to the 600-cell and other 4-polytopes</span> </div> </a> <ul id="toc-Relations_to_the_600-cell_and_other_4-polytopes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relations_to_other_uniform_polytopes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relations_to_other_uniform_polytopes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Relations to other uniform polytopes</span> </div> </a> <ul id="toc-Relations_to_other_uniform_polytopes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Appearances" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Appearances"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Appearances</span> </div> </a> <ul id="toc-Appearances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Regular 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 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Available in 19 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-19" 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">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%88%D6%82%D5%B2%D5%B2%D5%A1%D5%B1%D5%A5%D6%82_%D6%84%D5%BD%D5%A1%D5%B6%D5%A1%D5%B6%D5%AB%D5%BD%D5%BF" title="Ուղղաձեւ քսանանիստ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Ուղղաձեւ քսանանիստ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BA%D0%B0%D1%81%D0%B0%D1%8D%D0%B4%D1%80" title="Ікасаэдр – Belarusian" lang="be" hreflang="be" data-title="Ікасаэдр" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Korrap%C3%A4rane_ikosaeeder" title="Korrapärane ikosaeeder – Estonian" lang="et" hreflang="et" data-title="Korrapärane ikosaeeder" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%9D%B4%EC%8B%AD%EB%A9%B4%EC%B2%B4" title="정이십면체 – Korean" lang="ko" hreflang="ko" data-title="정이십면체" 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/%D4%BF%D5%A1%D5%B6%D5%B8%D5%B6%D5%A1%D5%BE%D5%B8%D6%80_%D6%84%D5%BD%D5%A1%D5%B6%D5%A1%D5%B6%D5%AB%D5%BD%D5%BF" title="Կանոնավոր քսանանիստ – Armenian" lang="hy" hreflang="hy" data-title="Կանոնավոր քսանանիստ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ikosahedron_beraturan" title="Ikosahedron beraturan – Indonesian" lang="id" hreflang="id" data-title="Ikosahedron beraturan" 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A9%D7%A8%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="עשרימון משוכלל – Hebrew" lang="he" hreflang="he" data-title="עשרימון משוכלל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Taisyklingas_ikosaedras" title="Taisyklingas ikosaedras – Lithuanian" lang="lt" hreflang="lt" data-title="Taisyklingas ikosaedras" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BA%D0%BE%D1%81%D0%B0%D0%B5%D0%B4%D0%B0%D1%80" title="Икосаедар – Macedonian" lang="mk" hreflang="mk" data-title="Икосаедар" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Regelmatig_twintigvlak" title="Regelmatig twintigvlak – Dutch" lang="nl" hreflang="nl" data-title="Regelmatig twintigvlak" 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/%E6%AD%A3%E4%BA%8C%E5%8D%81%E9%9D%A2%E4%BD%93" title="正二十面体 – Japanese" lang="ja" hreflang="ja" data-title="正二十面体" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dwudziesto%C5%9Bcian_foremny" title="Dwudziestościan foremny – Polish" lang="pl" hreflang="pl" data-title="Dwudziestościan foremny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B8%D0%BA%D0%BE%D1%81%D0%B0%D1%8D%D0%B4%D1%80" title="Правильный икосаэдр – Russian" lang="ru" hreflang="ru" data-title="Правильный икосаэдр" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ikosaedri_i_rregullt" title="Ikosaedri i rregullt – Albanian" lang="sq" hreflang="sq" data-title="Ikosaedri i rregullt" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BA%D0%BE%D1%81%D0%B0%D0%B5%D0%B4%D0%B0%D1%80" title="Икосаедар – Serbian" lang="sr" hreflang="sr" data-title="Икосаедар" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B0%E0%AF%81%E0%AE%AA%E0%AE%A4%E0%AF%81%E0%AE%AE%E0%AF%81%E0%AE%95_%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%95%E0%AE%AE%E0%AF%8D" title="இருபதுமுக முக்கோணகம் – Tamil" lang="ta" hreflang="ta" data-title="இருபதுமுக முக்கோணகம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D1%96%D0%BA%D0%BE%D1%81%D0%B0%D0%B5%D0%B4%D1%80" title="Правильний ікосаедр – Ukrainian" lang="uk" hreflang="uk" data-title="Правильний ікосаедр" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AD%A3%E5%BB%BF%E9%9D%A2%E9%AB%94" title="正廿面體 – Cantonese" lang="yue" hreflang="yue" data-title="正廿面體" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E4%BA%8C%E5%8D%81%E9%9D%A2%E9%AB%94" title="正二十面體 – Chinese" lang="zh" hreflang="zh" data-title="正二十面體" data-language-autonym="中文" data-language-local-name="Chinese" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Convex polyhedron with 20 triangular faces</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Regular icosahedron</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><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></span></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">Gyroelongated bipyramid</a><br /><a href="/wiki/Deltahedron" title="Deltahedron">Deltahedron</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Face_(geometry)" title="Face (geometry)">Faces</a></th><td class="infobox-data">20</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></th><td class="infobox-data">30</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></th><td class="infobox-data">12</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_configuration" title="Vertex configuration">Vertex configuration</a></th><td class="infobox-data"><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 12\times \left(3^{5}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>×</mo> <mrow> <mo>(</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\times \left(3^{5}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb321654cac013f374a779263eb5465aed8614ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.512ex; height:3.343ex;" alt="{\displaystyle 12\times \left(3^{5}\right)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">Icosahedral symmetry</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c84ca7860f66cd4ed8ecb07b4c5691f73c7365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.202ex; height:2.509ex;" alt="{\displaystyle I_{h}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dihedral_angle" title="Dihedral angle">Dihedral angle</a> (<a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>)</th><td class="infobox-data">138.190 (approximately)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></th><td class="infobox-data"><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Regular dodecahedron</a></td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="/wiki/Convex_set" title="Convex set">convex</a>,<br /><a href="/wiki/Composite_polyhedron" title="Composite polyhedron">composite</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e7dcc3"><a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">Net</a></th></tr><tr><td colspan="2" class="infobox-full-data"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Icosahedron_flat.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Icosahedron_flat.svg/280px-Icosahedron_flat.svg.png" decoding="async" width="280" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Icosahedron_flat.svg/420px-Icosahedron_flat.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Icosahedron_flat.svg/560px-Icosahedron_flat.svg.png 2x" data-file-width="236" data-file-height="120" /></a></span></td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the <b>regular icosahedron</b> (or simply <i>icosahedron</i>) is a convex <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> that can be constructed from <a href="/wiki/Pentagonal_antiprism" title="Pentagonal antiprism">pentagonal antiprism</a> by attaching two <a href="/wiki/Pentagonal_pyramid" title="Pentagonal pyramid">pentagonal pyramids</a> with <a href="/wiki/Regular_polygon" title="Regular polygon">regular faces</a> to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a> as its faces, 30 edges, and 12 vertices. It is an example of a <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a> and of a <a href="/wiki/Deltahedron" title="Deltahedron">deltahedron</a>. The icosahedral graph represents the <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">skeleton</a> of a regular icosahedron. </p><p>Many polyhedra are constructed from the regular icosahedron. For example, most of the <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedron</a> is constructed by <a href="/wiki/Faceting" title="Faceting">faceting</a>. Some of the <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a> can be constructed by removing the pentagonal pyramids. The regular icosahedron has many relations with other Platonic solids, one of them is the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> as its <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> and has the historical background on the comparison mensuration. It also has many relations with other <a href="/wiki/Polytope" title="Polytope">polytopes</a>. </p><p>The appearance of regular icosahedron can be found in nature, such as the virus with icosahedral-shaped <a href="/wiki/Capsid" title="Capsid">shells</a> and <a href="/wiki/Radiolarian" class="mw-redirect" title="Radiolarian">radiolarians</a>. Other applications of the regular icosahedron are the usage of its net in cartography, twenty-sided dice that may have been found in ancient times and <a href="/wiki/Role-playing_games" class="mw-redirect" title="Role-playing games">role-playing games</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=1" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The regular icosahedron can be constructed like other <a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramids</a>, started from a <a href="/wiki/Pentagonal_antiprism" title="Pentagonal antiprism">pentagonal antiprism</a> by attaching two <a href="/wiki/Pentagonal_pyramid" title="Pentagonal pyramid">pentagonal pyramids</a> with <a href="/wiki/Regular_polygon" title="Regular polygon">regular faces</a> to each of its faces. These pyramids cover the pentagonal faces, replacing them with five <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>, such that the resulting polyhedron has 20 equilateral triangles as its faces.<sup id="cite_ref-FOOTNOTESilvester2001[httpsbooksgooglecombooksidVtH_QG6scSUCpgPA141_140&ndash;141]Cundy1952_1-0" class="reference"><a href="#cite_note-FOOTNOTESilvester2001[httpsbooksgooglecombooksidVtH_QG6scSUCpgPA141_140&ndash;141]Cundy1952-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This process construction is known as the <a href="/wiki/Gyroelongation" class="mw-redirect" title="Gyroelongation">gyroelongation</a>.<sup id="cite_ref-FOOTNOTEBerman1971_2-0" class="reference"><a href="#cite_note-FOOTNOTEBerman1971-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedron-golden-rectangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Icosahedron-golden-rectangles.svg/220px-Icosahedron-golden-rectangles.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Icosahedron-golden-rectangles.svg/330px-Icosahedron-golden-rectangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Icosahedron-golden-rectangles.svg/440px-Icosahedron-golden-rectangles.svg.png 2x" data-file-width="500" data-file-height="463" /></a><figcaption>Three mutually perpendicular <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> rectangles, with edges connecting their corners, form a regular icosahedron.</figcaption></figure> <p>Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage70mode1upviewtheater_70]_3-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage70mode1upviewtheater_70]-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Because of the constructions above, the regular icosahedron is <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>, a family of polyhedra with <a href="/wiki/Regular_polygon" title="Regular polygon">regular faces</a>. A polyhedron with only equilateral triangles as faces is called a <a href="/wiki/Deltahedron" title="Deltahedron">deltahedron</a>. There are only eight different convex deltahedra, one of which is the regular icosahedron.<sup id="cite_ref-FOOTNOTEShavinina2013[httpsbooksgooglecombooksidJcPd_JRc4FgCpgPA333_333]Cundy1952_4-0" class="reference"><a href="#cite_note-FOOTNOTEShavinina2013[httpsbooksgooglecombooksidJcPd_JRc4FgCpgPA333_333]Cundy1952-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The regular icosahedron can also be constructed starting from a <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>. All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles. This process is known as <a href="/wiki/Snub_(geometry)" title="Snub (geometry)">snub</a>, and the regular icosahedron is also known as <b>snub octahedron</b>.<sup id="cite_ref-FOOTNOTEKappraff1991[httpsbooksgooglecombooksidtz76s0ZGFiQCpgPA475_475]_5-0" class="reference"><a href="#cite_note-FOOTNOTEKappraff1991[httpsbooksgooglecombooksidtz76s0ZGFiQCpgPA475_475]-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>One possible system of <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinate</a> for the vertices of a regular icosahedron, giving the edge length 2, is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>φ</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mi>φ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0098f1a6086071cd31b104a58ba5ecc890e91d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.891ex; height:2.843ex;" alt="{\displaystyle \left(0,\pm 1,\pm \varphi \right),\left(\pm 1,\pm \varphi ,0\right),\left(\pm \varphi ,0,\pm 1\right),}"></span> where <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 \varphi =(1+{\sqrt {5}})/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =(1+{\sqrt {5}})/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f409bafc8166035e6535ed6bb1a12ccb2d97d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.854ex; height:3.009ex;" alt="{\displaystyle \varphi =(1+{\sqrt {5}})/2}"></span> denotes the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>.<sup id="cite_ref-FOOTNOTESteebHardyTanski2012[httpsbooksgooglecombooksidUdI7DQAAQBAJpgPA211_211]_6-0" class="reference"><a href="#cite_note-FOOTNOTESteebHardyTanski2012[httpsbooksgooglecombooksidUdI7DQAAQBAJpgPA211_211]-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mensuration">Mensuration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=3" title="Edit section: Mensuration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span class="mw-3d-wrapper" data-label="3D"><a href="/wiki/File:Regular_icosahedron.stl" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Regular_icosahedron.stl/220px-Regular_icosahedron.stl.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Regular_icosahedron.stl/330px-Regular_icosahedron.stl.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Regular_icosahedron.stl/440px-Regular_icosahedron.stl.png 2x" data-file-width="5120" data-file-height="2880" /></a></span><figcaption>3D model of a regular icosahedron</figcaption></figure> <p>The <a href="/wiki/Insphere" class="mw-redirect" title="Insphere">insphere</a> of a convex polyhedron is a sphere inside the polyhedron, touching every face. The <a href="/wiki/Circumsphere" class="mw-redirect" title="Circumsphere">circumsphere</a> of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The <a href="/wiki/Midsphere" title="Midsphere">midsphere</a> of a convex polyhedron is a sphere tangent to every edge. Therefore, given that the edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> of a regular icosahedron, the radius of insphere (inradius) <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 r_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e177223a0b94e134c613f3e13b02a6050229c1f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.109ex; height:2.009ex;" alt="{\displaystyle r_{I}}"></span>, the radius of circumsphere (circumradius) <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 r_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4045733e6ea6c11a3061b638c8c046cb43f06a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.53ex; height:2.009ex;" alt="{\displaystyle r_{C}}"></span>, and the radius of midsphere (midradius) <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 r_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24df07649f274efde2fe766dbfb26e5c4d51409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.008ex; height:2.009ex;" alt="{\displaystyle r_{M}}"></span> are, respectively:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{I}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}\approx 0.756a,\qquad r_{C}={\frac {\sqrt {\varphi ^{2}+1}}{2}}a\approx 0.951a,\qquad r_{M}={\frac {\varphi }{2}}a\approx 0.809a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>0.756</mn> <mi>a</mi> <mo>,</mo> <mspace width="2em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mo>≈</mo> <mn>0.951</mn> <mi>a</mi> <mo>,</mo> <mspace width="2em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mo>≈</mo> <mn>0.809</mn> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{I}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}\approx 0.756a,\qquad r_{C}={\frac {\sqrt {\varphi ^{2}+1}}{2}}a\approx 0.951a,\qquad r_{M}={\frac {\varphi }{2}}a\approx 0.809a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de886bc95c8580cf6411a31355b333c674ee961" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:77.472ex; height:7.176ex;" alt="{\displaystyle r_{I}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}\approx 0.756a,\qquad r_{C}={\frac {\sqrt {\varphi ^{2}+1}}{2}}a\approx 0.951a,\qquad r_{M}={\frac {\varphi }{2}}a\approx 0.809a.}"></span> </p><p>The <a href="/wiki/Surface_area" title="Surface area">surface area</a> of a polyhedron is the sum of the areas of its faces. Therefore, the surface area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80b1b4d0e79363eda23bd4ddf0c26e84f071ffe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.552ex; height:2.843ex;" alt="{\displaystyle (A)}"></span> of a regular icosahedron is 20 times that of each of its equilateral triangle faces. The volume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb43b397cf61f76d7438c2835e3ed80c4b235395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.597ex; height:2.843ex;" alt="{\displaystyle (V)}"></span> of a regular icosahedron can be obtained as 20 times that of a pyramid whose base is one of its faces and whose apex is the icosahedron's center; or as the sum of two uniform <a href="/wiki/Pentagonal_pyramid" title="Pentagonal pyramid">pentagonal pyramids</a> and a <a href="/wiki/Pentagonal_antiprism" title="Pentagonal antiprism">pentagonal antiprism</a>. The expressions of both are:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mn>8.660</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>6</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≈</mo> <mn>2.182</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb587f8f0a649f96bc93123ce1f7ed60a3c3e4b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.813ex; height:5.676ex;" alt="{\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.}"></span> A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by <a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Hero</a>, <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus</a>, and <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>, among others.<sup id="cite_ref-FOOTNOTEHerz-Fischler2013[httpsbooksgooglecombooksidaYjXZJwLARQCpgPA138_138–140]_9-0" class="reference"><a href="#cite_note-FOOTNOTEHerz-Fischler2013[httpsbooksgooglecombooksidaYjXZJwLARQCpgPA138_138–140]-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a> discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas.<sup id="cite_ref-FOOTNOTESimmons2007[httpsbooksgooglecombooksid3KOst4Mon90CpgPA50_50]_10-0" class="reference"><a href="#cite_note-FOOTNOTESimmons2007[httpsbooksgooglecombooksid3KOst4Mon90CpgPA50_50]-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Both volumes have formulas involving the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>, but taken to different powers.<sup id="cite_ref-FOOTNOTESutton2002[httpsbooksgooglecombooksidvgo7bTxDmIsCpgPA55_55]_11-0" class="reference"><a href="#cite_note-FOOTNOTESutton2002[httpsbooksgooglecombooksidvgo7bTxDmIsCpgPA55_55]-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces is approximately 38.2°. The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is 100.8°, and the dihedral angle of a pentagonal pyramid between the same faces is 37.4°. Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is 37.4° + 100.8° = 138.2°.<sup id="cite_ref-FOOTNOTEJohnson1966See_table_II,_line_4.MacLean2007[httpsbooksgooglecombooksidvINuAwAAQBAJpgPA44_43&ndash;44]_13-0" class="reference"><a href="#cite_note-FOOTNOTEJohnson1966See_table_II,_line_4.MacLean2007[httpsbooksgooglecombooksidvINuAwAAQBAJpgPA44_43&ndash;44]-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=4" title="Edit section: Symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">Icosahedral symmetry</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_symmetry_group_ih.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/220px-Sphere_symmetry_group_ih.png" decoding="async" width="220" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/330px-Sphere_symmetry_group_ih.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/440px-Sphere_symmetry_group_ih.png 2x" data-file-width="671" data-file-height="617" /></a><figcaption>Full <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">Icosahedral symmetry</a> has 15 mirror planes (seen as cyan <a href="/wiki/Great_circle" title="Great circle">great circles</a> on this sphere) meeting at order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /5,\pi /3,\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo>,</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>,</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /5,\pi /3,\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a32542203d8f62ff8191ae4b586e7351c273d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.039ex; height:2.843ex;" alt="{\displaystyle \pi /5,\pi /3,\pi /2}"></span> angles, dividing a sphere into 120 triangle <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domains</a>. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points.</figcaption></figure> <p>The rotational <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of the regular icosahedron is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> on five letters. This non-<a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Simple_group" title="Simple group">simple group</a> is the only non-trivial <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> on five letters. Since the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of the general <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a> is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> uses this simple fact, and <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The full symmetry group of the icosahedron (including reflections) is known as the <a href="/wiki/Full_icosahedral_group" class="mw-redirect" title="Full icosahedral group">full icosahedral group</a>. It is isomorphic to the product of the rotational symmetry group and the group <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 C_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec545f7870665e1028b7492746848d149878808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}"></span> of size two, which is generated by the reflection through the center of the icosahedron. </p> <div class="mw-heading mw-heading2"><h2 id="Icosahedral_graph">Icosahedral graph</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=5" title="Edit section: Icosahedral graph"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Icosahedron_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Icosahedron_graph.svg/220px-Icosahedron_graph.svg.png" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Icosahedron_graph.svg/330px-Icosahedron_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Icosahedron_graph.svg/440px-Icosahedron_graph.svg.png 2x" data-file-width="625" data-file-height="584" /></a><figcaption>Icosahedral graph</figcaption></figure> <p>Every <a href="/wiki/Platonic_graph" class="mw-redirect" title="Platonic graph">Platonic graph</a>, including the <b>icosahedral graph</b>, is a <a href="/wiki/Polyhedral_graph" title="Polyhedral graph">polyhedral graph</a>. This means that they are <a href="/wiki/Planar_graph" title="Planar graph">planar graphs</a>, graphs that can be drawn in the plane without crossing its edges; and they are <a href="/wiki/K-vertex-connected_graph" title="K-vertex-connected graph">3-vertex-connected</a>, meaning that the removal of any two of its vertices leaves a connected subgraph. According to <a href="/wiki/Steinitz_theorem" class="mw-redirect" title="Steinitz theorem">Steinitz theorem</a>, the icosahedral graph endowed with these heretofore properties represents the <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">skeleton</a> of a regular icosahedron.<sup id="cite_ref-FOOTNOTEBickle2020[httpsbooksgooglecombooksid2sbVDwAAQBAJpgPA147_147]_15-0" class="reference"><a href="#cite_note-FOOTNOTEBickle2020[httpsbooksgooglecombooksid2sbVDwAAQBAJpgPA147_147]-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The icosahedral graph is <a href="/wiki/Hamiltonian_graph" class="mw-redirect" title="Hamiltonian graph">Hamiltonian</a>, meaning that it contains a Hamiltonian cycle, or a cycle that visits each vertex exactly once.<sup id="cite_ref-FOOTNOTEHopkins2004_16-0" class="reference"><a href="#cite_note-FOOTNOTEHopkins2004-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_polyhedra">Related polyhedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=6" title="Edit section: Related polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_other_Platonic_solids">In other Platonic solids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=7" title="Edit section: In other Platonic solids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aside from comparing the mensuration between the regular icosahedron and regular dodecahedron, they are <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> to each other. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.<sup id="cite_ref-FOOTNOTEHerrmannSally2013[httpsbooksgooglecombooksidb2fjR81h6yECpgPA257_257]_17-0" class="reference"><a href="#cite_note-FOOTNOTEHerrmannSally2013[httpsbooksgooglecombooksidb2fjR81h6yECpgPA257_257]-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>An icosahedron can be inscribed in an <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two <a href="/wiki/Golden_section" class="mw-redirect" title="Golden section">golden sections</a>. Because the golden sections are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.<sup id="cite_ref-FOOTNOTECoxeterdu_ValFlatherPetrie19384_18-0" class="reference"><a href="#cite_note-FOOTNOTECoxeterdu_ValFlatherPetrie19384-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>An icosahedron of edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\varphi }}\approx 0.618}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> <mo>≈</mo> <mn>0.618</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\varphi }}\approx 0.618}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63071ef02eb10f7182f251663bdc2a19b491c273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.306ex; height:3.676ex;" alt="{\textstyle {\frac {1}{\varphi }}\approx 0.618}"></span> can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges.<sup id="cite_ref-FOOTNOTEBorovik20068–9§5._How_to_draw_an_icosahedron_on_a_blackboard_19-0" class="reference"><a href="#cite_note-FOOTNOTEBorovik20068–9§5._How_to_draw_an_icosahedron_on_a_blackboard-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Because there are five times as many icosahedron edges as cube faces, there are five ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Stellation">Stellation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=8" title="Edit section: Stellation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The icosahedron has a large number of <a href="/wiki/Stellation" title="Stellation">stellations</a>. <a href="#CITEREFCoxeterdu_ValFlatherPetrie1938">Coxeter et al. (1938)</a> stated 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">Kepler–Poinsot polyhedron</a>. Three are <a href="/wiki/Polyhedral_compound#Regular_compounds" class="mw-redirect" title="Polyhedral compound">regular compound polyhedra</a>.<sup id="cite_ref-FOOTNOTECoxeterdu_ValFlatherPetrie19388&ndash;26_21-0" class="reference"><a href="#cite_note-FOOTNOTECoxeterdu_ValFlatherPetrie19388&ndash;26-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <caption>21 of 59 stellations </caption> <tbody><tr> <td rowspan="3" width="200"><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/200px-Stellation_diagram_of_icosahedron.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stellation_diagram_of_icosahedron.svg/300px-Stellation_diagram_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stellation_diagram_of_icosahedron.svg/400px-Stellation_diagram_of_icosahedron.svg.png 2x" data-file-width="1120" data-file-height="1120" /></a></span><br />The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this <a href="/wiki/Stellation_diagram" title="Stellation diagram">stellation diagram</a> of the intersections in a single plane. </td> <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/50px-Zeroth_stellation_of_icosahedron.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Zeroth_stellation_of_icosahedron.svg/75px-Zeroth_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Zeroth_stellation_of_icosahedron.svg/100px-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/50px-First_stellation_of_icosahedron.png" decoding="async" width="50" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/First_stellation_of_icosahedron.png/75px-First_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/First_stellation_of_icosahedron.png/100px-First_stellation_of_icosahedron.png 2x" data-file-width="920" data-file-height="900" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Second_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Second_stellation_of_icosahedron.png/50px-Second_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Second_stellation_of_icosahedron.png/75px-Second_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Second_stellation_of_icosahedron.png/100px-Second_stellation_of_icosahedron.png 2x" data-file-width="1000" data-file-height="1000" /></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/50px-Third_stellation_of_icosahedron.svg.png" decoding="async" width="50" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Third_stellation_of_icosahedron.svg/75px-Third_stellation_of_icosahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Third_stellation_of_icosahedron.svg/100px-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:Fourth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Fourth_stellation_of_icosahedron.png/50px-Fourth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Fourth_stellation_of_icosahedron.png/75px-Fourth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Fourth_stellation_of_icosahedron.png/100px-Fourth_stellation_of_icosahedron.png 2x" data-file-width="899" data-file-height="900" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fifth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Fifth_stellation_of_icosahedron.png/50px-Fifth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Fifth_stellation_of_icosahedron.png/75px-Fifth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Fifth_stellation_of_icosahedron.png/100px-Fifth_stellation_of_icosahedron.png 2x" data-file-width="919" data-file-height="921" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sixth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Sixth_stellation_of_icosahedron.png/50px-Sixth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Sixth_stellation_of_icosahedron.png/75px-Sixth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Sixth_stellation_of_icosahedron.png/100px-Sixth_stellation_of_icosahedron.png 2x" data-file-width="900" data-file-height="900" /></a></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Seventh_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Seventh_stellation_of_icosahedron.png/50px-Seventh_stellation_of_icosahedron.png" decoding="async" width="50" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Seventh_stellation_of_icosahedron.png/75px-Seventh_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Seventh_stellation_of_icosahedron.png/100px-Seventh_stellation_of_icosahedron.png 2x" data-file-width="920" data-file-height="900" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Eighth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Eighth_stellation_of_icosahedron.png/50px-Eighth_stellation_of_icosahedron.png" decoding="async" width="50" height="47" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Eighth_stellation_of_icosahedron.png/75px-Eighth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Eighth_stellation_of_icosahedron.png/100px-Eighth_stellation_of_icosahedron.png 2x" data-file-width="950" data-file-height="899" /></a></span> </td> <td><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/50px-Ninth_stellation_of_icosahedron.png" decoding="async" width="50" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Ninth_stellation_of_icosahedron.png/75px-Ninth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Ninth_stellation_of_icosahedron.png/100px-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:Tenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Tenth_stellation_of_icosahedron.png/50px-Tenth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Tenth_stellation_of_icosahedron.png/75px-Tenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Tenth_stellation_of_icosahedron.png/100px-Tenth_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:Eleventh_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Eleventh_stellation_of_icosahedron.png/50px-Eleventh_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Eleventh_stellation_of_icosahedron.png/75px-Eleventh_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Eleventh_stellation_of_icosahedron.png/100px-Eleventh_stellation_of_icosahedron.png 2x" data-file-width="600" data-file-height="599" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Twelfth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Twelfth_stellation_of_icosahedron.png/50px-Twelfth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Twelfth_stellation_of_icosahedron.png/75px-Twelfth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Twelfth_stellation_of_icosahedron.png/100px-Twelfth_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:Thirteenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Thirteenth_stellation_of_icosahedron.png/50px-Thirteenth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Thirteenth_stellation_of_icosahedron.png/75px-Thirteenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Thirteenth_stellation_of_icosahedron.png/100px-Thirteenth_stellation_of_icosahedron.png 2x" data-file-width="899" data-file-height="900" /></a></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Fourteenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Fourteenth_stellation_of_icosahedron.png/50px-Fourteenth_stellation_of_icosahedron.png" decoding="async" width="50" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Fourteenth_stellation_of_icosahedron.png/75px-Fourteenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Fourteenth_stellation_of_icosahedron.png/100px-Fourteenth_stellation_of_icosahedron.png 2x" data-file-width="921" data-file-height="901" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fifteenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Fifteenth_stellation_of_icosahedron.png/50px-Fifteenth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Fifteenth_stellation_of_icosahedron.png/75px-Fifteenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Fifteenth_stellation_of_icosahedron.png/100px-Fifteenth_stellation_of_icosahedron.png 2x" data-file-width="901" data-file-height="900" /></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/50px-Sixteenth_stellation_of_icosahedron.png" decoding="async" width="50" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/75px-Sixteenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Sixteenth_stellation_of_icosahedron.png/100px-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:Seventeenth_stellation_of_icosahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Seventeenth_stellation_of_icosahedron.png/50px-Seventeenth_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Seventeenth_stellation_of_icosahedron.png/75px-Seventeenth_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Seventeenth_stellation_of_icosahedron.png/100px-Seventeenth_stellation_of_icosahedron.png 2x" data-file-width="909" 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/50px-First_compound_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_compound_stellation_of_icosahedron.png/75px-First_compound_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/First_compound_stellation_of_icosahedron.png/100px-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/50px-Second_compound_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Second_compound_stellation_of_icosahedron.png/75px-Second_compound_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Second_compound_stellation_of_icosahedron.png/100px-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/50px-Third_compound_stellation_of_icosahedron.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Third_compound_stellation_of_icosahedron.png/75px-Third_compound_stellation_of_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Third_compound_stellation_of_icosahedron.png/100px-Third_compound_stellation_of_icosahedron.png 2x" data-file-width="899" data-file-height="899" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Facetings">Facetings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=9" title="Edit section: Facetings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:129px;max-width:129px"><div class="thumbimage" style="height:134px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Great_dodecahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/127px-Great_dodecahedron.png" decoding="async" width="127" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/191px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/254px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958" /></a></span></div></div><div class="tsingle" style="width:129px;max-width:129px"><div class="thumbimage" style="height:134px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Small_stellated_dodecahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/127px-Small_stellated_dodecahedron.png" decoding="async" width="127" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/191px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/254px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953" /></a></span></div></div><div class="tsingle" style="width:128px;max-width:128px"><div class="thumbimage" style="height:134px;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/126px-Great_icosahedron.png" decoding="async" width="126" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/189px-Great_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Great_icosahedron.png/252px-Great_icosahedron.png 2x" data-file-width="1830" data-file-height="1954" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The <a href="/wiki/Great_dodecahedron" title="Great dodecahedron">great dodecahedron</a>, <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a>, and the <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a>.</div></div></div></div> <p>The <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a>, <a href="/wiki/Great_dodecahedron" title="Great dodecahedron">great dodecahedron</a>, and <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a> are three <a href="/wiki/Faceting" title="Faceting">facetings</a> of the regular icosahedron. They share the same <a href="/wiki/Vertex_arrangement" title="Vertex arrangement">vertex arrangement</a>. They all have 30 edges. The regular icosahedron and great dodecahedron share the same <a href="/wiki/Edge_arrangement" class="mw-redirect" title="Edge arrangement">edge arrangement</a> but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles). </p> <div class="mw-heading mw-heading3"><h3 id="Diminishment">Diminishment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=10" title="Edit section: Diminishment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solid</a> is a polyhedron whose faces are all regular, but which is not uniform. This means the Johnson solids do not include the <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a>, the <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a>, the <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a>, or the <a href="/wiki/Antiprism" title="Antiprism">antiprisms</a>. Some of them are constructed involving the removal of the part of a regular icosahedron, a process known as <i>diminishment</i>. They are <a href="/wiki/Gyroelongated_pentagonal_pyramid" title="Gyroelongated pentagonal pyramid">gyroelongated pentagonal pyramid</a>, <a href="/wiki/Metabidiminished_icosahedron" title="Metabidiminished icosahedron">metabidiminished icosahedron</a>, and <a href="/wiki/Tridiminished_icosahedron" title="Tridiminished icosahedron">tridiminished icosahedron</a>, which remove one, two, and three pentagonal pyramids from the icosahedron, respectively.<sup id="cite_ref-FOOTNOTEBerman1971_2-1" class="reference"><a href="#cite_note-FOOTNOTEBerman1971-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The similar <a href="/wiki/Dissected_regular_icosahedron" class="mw-redirect" title="Dissected regular icosahedron">dissected regular icosahedron</a> has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Relations_to_the_600-cell_and_other_4-polytopes">Relations to the 600-cell and other 4-polytopes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=11" title="Edit section: Relations to the 600-cell and other 4-polytopes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The icosahedron is the <a href="/wiki/Four-dimensional_space#Dimensional_analogy" title="Four-dimensional space">dimensional analogue</a> of the <a href="/wiki/600-cell" title="600-cell">600-cell</a>, a <a href="/wiki/Regular_4-polytope#Regular_convex_4-polytopes" title="Regular 4-polytope">regular 4-dimensional polytope</a>. The 600-cell has icosahedral <a href="/wiki/600-cell#Polyhedral_sections" title="600-cell">cross sections</a> of two sizes, and each of its 120 vertices is an <a href="/wiki/Icosahedral_pyramid" title="Icosahedral pyramid">icosahedral pyramid</a>; the icosahedron is the <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> of the 600-cell. </p><p>The unit-radius 600-cell has tetrahedral cells of edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399b5a609eb41c83ddf113a479292be45d4dc5ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.911ex; height:3.676ex;" alt="{\textstyle {\frac {1}{\varphi }}}"></span>, 20 of which meet at each vertex to form an icosahedral pyramid (a <a href="/wiki/4-pyramid" class="mw-redirect" title="4-pyramid">4-pyramid</a> with an icosahedron as its base). Thus the 600-cell contains 120 icosahedra of edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399b5a609eb41c83ddf113a479292be45d4dc5ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.911ex; height:3.676ex;" alt="{\textstyle {\frac {1}{\varphi }}}"></span>. The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as <a href="/wiki/600-cell#Cell_clusters" title="600-cell">interior features</a> formed by its unit-length <a href="/wiki/600-cell#Golden_chords" title="600-cell">chords</a>. In the unit-radius <a href="/wiki/120-cell" title="120-cell">120-cell</a> (another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron, and in a cube). </p><p>A semiregular 4-polytope, the <a href="/wiki/Snub_24-cell" title="Snub 24-cell">snub 24-cell</a>, has icosahedral cells. </p> <div class="mw-heading mw-heading3"><h3 id="Relations_to_other_uniform_polytopes">Relations to other uniform polytopes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=12" title="Edit section: Relations to other uniform polytopes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As mentioned above, the regular icosahedron is unique among the <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a> in possessing a <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> is approximately <span class="nowrap"><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 138.19^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>138.19</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 138.19^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed0f0691d29c46c3e87948db8120ce873847b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.513ex; height:2.343ex;" alt="{\displaystyle 138.19^{\circ }}"></span>.</span> Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive <a href="/wiki/Defect_(geometry)" class="mw-redirect" title="Defect (geometry)">defect</a> for folding in three dimensions, icosahedra cannot be used as the <a href="/wiki/Cell_(geometry)" class="mw-redirect" title="Cell (geometry)">cells</a> of a convex regular <a href="/wiki/Polychoron" class="mw-redirect" title="Polychoron">polychoron</a> because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex <a href="/wiki/Polytope" title="Polytope">polytope</a> in <i>n</i> dimensions, at least three <a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">facets</a> must meet at a <a href="/wiki/Peak_(geometry)" class="mw-redirect" title="Peak (geometry)">peak</a> and leave a positive defect for folding in <i>n</i>-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the <a href="/wiki/Snub_24-cell" title="Snub 24-cell">snub 24-cell</a>), just as hexagons can be used as faces in semi-regular polyhedra (for example the <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a>). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the <a href="/wiki/Icosahedral_120-cell" title="Icosahedral 120-cell">icosahedral 120-cell</a>, one of the ten <a href="/wiki/Schl%C3%A4fli%E2%80%93Hess_polychoron" class="mw-redirect" title="Schläfli–Hess polychoron">non-convex regular polychora</a>. </p><p>There are distortions of the icosahedron that, while no longer regular, are nevertheless <a href="/wiki/Vertex-uniform" class="mw-redirect" title="Vertex-uniform">vertex-uniform</a>. These are <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> under the same <a href="/wiki/Rotation" title="Rotation">rotations</a> as the tetrahedron, and are somewhat analogous to the <a href="/wiki/Snub_cube" title="Snub cube">snub cube</a> and <a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a>, including some forms which are <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a> and some with <span class="nowrap"><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 T_{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eaacb63fcdf3a344175b45b3bab7910f8fc3e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.537ex; height:2.509ex;" alt="{\displaystyle T_{h}}"></span>-</span>symmetry, i.e. have different planes of symmetry from the tetrahedron. </p> <div class="mw-heading mw-heading2"><h2 id="Appearances">Appearances</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=13" title="Edit section: Appearances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:141px;max-width:141px"><div class="thumbimage" style="height:149px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Twenty-sided_die_(icosahedron)_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Twenty-sided_die_%28icosahedron%29_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg/139px-Twenty-sided_die_%28icosahedron%29_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg" decoding="async" width="139" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Twenty-sided_die_%28icosahedron%29_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg/209px-Twenty-sided_die_%28icosahedron%29_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Twenty-sided_die_%28icosahedron%29_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg/278px-Twenty-sided_die_%28icosahedron%29_with_faces_inscribed_with_Greek_letters_MET_10.130.1158_001.jpg 2x" data-file-width="1488" data-file-height="1596" /></a></span></div><div class="thumbcaption">Twenty-sided dice from Ptolemaic of Egypt, inscribed with Greek letters at the faces.</div></div><div class="tsingle" style="width:147px;max-width:147px"><div class="thumbimage" style="height:149px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:ScatDice.JPG" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ScatDice.JPG/145px-ScatDice.JPG" decoding="async" width="145" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ScatDice.JPG/218px-ScatDice.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/ScatDice.JPG/290px-ScatDice.JPG 2x" data-file-width="322" data-file-height="331" /></a></span></div><div class="thumbcaption">The <i>Scattergories</i> twenty-sided die, excluding the six letters Q, U, V, X, Y, and Z.</div></div></div></div></div> <p>Dice are the most common objects using different polyhedra, one of them being the regular icosahedron. The twenty-sided die was found in many ancient times. One example is the die from the Ptolemaic of Egypt, which later used Greek letters inscribed on the faces in the period of Greece and Rome.<sup id="cite_ref-FOOTNOTESmith1958[httpsbooksgooglecombooksiduTytJGnTf1kCpgPA295_295]Minas-Nerpel2007_22-0" class="reference"><a href="#cite_note-FOOTNOTESmith1958[httpsbooksgooglecombooksiduTytJGnTf1kCpgPA295_295]Minas-Nerpel2007-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Another example was found in the treasure of <a href="/wiki/Tipu_Sultan" title="Tipu Sultan">Tipu Sultan</a>, which was made out of gold and with numbers written on each face.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage4mode1upviewtheater_4]_23-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage4mode1upviewtheater_4]-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In several <a href="/wiki/Roleplaying_game" class="mw-redirect" title="Roleplaying game">roleplaying games</a>, such as <i><a href="/wiki/Dungeons_%26_Dragons" title="Dungeons & Dragons">Dungeons & Dragons</a></i>, the twenty-sided die (labeled as <a href="/wiki/Dice#Polyhedral_dice" title="Dice">d20</a>) is commonly used in determining success or failure of an action. It may be numbered from "0" to "9" twice, in which form it usually serves as a ten-sided die (<a href="/wiki/Dice#Non-cubical_dice" title="Dice">d10</a>); most modern versions are labeled from "1" to "20".<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <i><a href="/wiki/Scattergories" title="Scattergories">Scattergories</a></i> is another board game in which the player names the category entires on a card within a given set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.<sup id="cite_ref-FOOTNOTEFlanaganGregory2015[httpsbooksgooglecombooksidhViuEAAAQBAJpgPA85_85]_25-0" class="reference"><a href="#cite_note-FOOTNOTEFlanaganGregory2015[httpsbooksgooglecombooksidhViuEAAAQBAJpgPA85_85]-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:342px;max-width:342px"><div class="trow"><div class="tsingle" style="width:102px;max-width:102px"><div class="thumbimage" style="height:110px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Circogonia_icosahedra.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Circogonia_icosahedra.jpg/100px-Circogonia_icosahedra.jpg" decoding="async" width="100" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Circogonia_icosahedra.jpg/150px-Circogonia_icosahedra.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Circogonia_icosahedra.jpg/200px-Circogonia_icosahedra.jpg 2x" data-file-width="1170" data-file-height="1298" /></a></span></div><div class="thumbcaption">The <a href="/wiki/Radiolarian" class="mw-redirect" title="Radiolarian">radiolarian</a> <i>Circogonia icosahedra</i></div></div><div class="tsingle" style="width:236px;max-width:236px"><div class="thumbimage" style="height:110px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Dymaxion_projection.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Dymaxion_projection.png/234px-Dymaxion_projection.png" decoding="async" width="234" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Dymaxion_projection.png/351px-Dymaxion_projection.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Dymaxion_projection.png/468px-Dymaxion_projection.png 2x" data-file-width="2180" data-file-height="1030" /></a></span></div><div class="thumbcaption"><a href="/wiki/Dymaxion_map" title="Dymaxion map">Dymaxion map</a>, created by the net of a regular icosahedron</div></div></div></div></div> <p>The regular icosahedron may also appear in many fields of science as follows: </p> <ul><li>In <a href="/wiki/Virology" title="Virology">virology</a>, <a href="/wiki/Herpesviridae" title="Herpesviridae">herpes virus</a> have icosahedral <a href="/wiki/Capsid" title="Capsid">shells</a>. The outer protein shell of <a href="/wiki/HIV" title="HIV">HIV</a> is enclosed in a regular icosahedron, as is the head of a typical <a href="/wiki/Myovirus" class="mw-redirect" title="Myovirus">myovirus</a>.<sup id="cite_ref-FOOTNOTEStraussStrauss200835&ndash;62_26-0" class="reference"><a href="#cite_note-FOOTNOTEStraussStrauss200835&ndash;62-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Several species of <a href="/wiki/Radiolarian" class="mw-redirect" title="Radiolarian">radiolarians</a> discovered by <a href="/wiki/Ernst_Haeckel" title="Ernst Haeckel">Ernst Haeckel</a>, described its shells as the like-shaped various regular polyhedra; one of which is <i>Circogonia icosahedra</i>, whose skeleton is shaped like a regular icosahedron.<sup id="cite_ref-FOOTNOTEHaeckel1904Cromwell1997[httpsarchiveorgdetailspolyhedra0000crompage6mode1upviewtheater_6]_27-0" class="reference"><a href="#cite_note-FOOTNOTEHaeckel1904Cromwell1997[httpsarchiveorgdetailspolyhedra0000crompage6mode1upviewtheater_6]-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li>In chemistry, the <a href="/wiki/Closo_cluster" class="mw-redirect" title="Closo cluster">closo</a>-<a href="/wiki/Carboranes" class="mw-redirect" title="Carboranes">carboranes</a> are <a href="/wiki/Chemical_compound" title="Chemical compound">compounds</a> with a shape resembling the regular icosahedron.<sup id="cite_ref-FOOTNOTESpokoyny2013_28-0" class="reference"><a href="#cite_note-FOOTNOTESpokoyny2013-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Crystal_twinning" title="Crystal twinning">crystal twinning</a> with <a href="/wiki/Icosahedral_twins" title="Icosahedral twins">icosahedral shapes</a> also occurs in crystals, especially <a href="/wiki/Nanoparticle" title="Nanoparticle">nanoparticles</a>.<sup id="cite_ref-FOOTNOTEHofmeister2004_29-0" class="reference"><a href="#cite_note-FOOTNOTEHofmeister2004-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Many <a href="/wiki/Crystal_structure_of_boron-rich_metal_borides" title="Crystal structure of boron-rich metal borides">borides</a> and <a href="/wiki/Allotropes_of_boron" title="Allotropes of boron">allotropes of boron</a> such as <a href="/wiki/Allotropes_of_boron#α-rhombohedral_boron" title="Allotropes of boron">α-</a> and <a href="/wiki/Allotropes_of_boron#β-rhombohedral_boron" title="Allotropes of boron">β-rhombohedral</a> contain boron B<sub>12</sub> icosahedron as a basic structure unit.<sup id="cite_ref-FOOTNOTEDronskowskiKikkawaStein2017[httpsbooksgooglecombookside0VBDwAAQBAJpgPA436_435&ndash;436]_30-0" class="reference"><a href="#cite_note-FOOTNOTEDronskowskiKikkawaStein2017[httpsbooksgooglecombookside0VBDwAAQBAJpgPA436_435&ndash;436]-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li> <li>In cartography, <a href="/wiki/R._Buckminster_Fuller" class="mw-redirect" title="R. Buckminster Fuller">R. Buckminster Fuller</a> used the net of a regular icosahedron to create a map known as <a href="/wiki/Dymaxion_map" title="Dymaxion map">Dymaxion map</a>, by subdividing the net into triangles, followed by calculating the grid on the Earth's surface, and transferring the results from the sphere to the polyhedron. This projection was created during the time that Fuller realized that the <a href="/wiki/Greenland" title="Greenland">Greenland</a> is smaller than <a href="/wiki/South_America" title="South America">South America</a>.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage7mode1upviewtheater_7]_31-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage7mode1upviewtheater_7]-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></li> <li>In the <a href="/wiki/Thomson_problem" title="Thomson problem">Thomson problem</a>, concerning the minimum-energy configuration of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> charged particles on a sphere, and for the <a href="/wiki/Tammes_problem" title="Tammes problem">Tammes problem</a> of constructing a <a href="/wiki/Spherical_code" title="Spherical code">spherical code</a> maximizing the smallest distance among the points, the minimum solution known for <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 n=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38fb7dfd38b5db65d78bb6f2c1bf9e6e6cd07c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=12}"></span> places the points at the vertices of a regular icosahedron, <a href="/wiki/Circumscribed_sphere" title="Circumscribed sphere">inscribed in a sphere</a>. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.<sup id="cite_ref-FOOTNOTEWhyte1952_32-0" class="reference"><a href="#cite_note-FOOTNOTEWhyte1952-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:151px;max-width:151px"><div class="thumbimage" style="height:159px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Kepler_Icosahedron_Water.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler_Icosahedron_Water.jpg/149px-Kepler_Icosahedron_Water.jpg" decoding="async" width="149" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler_Icosahedron_Water.jpg/224px-Kepler_Icosahedron_Water.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Kepler_Icosahedron_Water.jpg/298px-Kepler_Icosahedron_Water.jpg 2x" data-file-width="306" data-file-height="328" /></a></span></div><div class="thumbcaption">Sketch of a regular icosahedron by Johannes Kepler</div></div><div class="tsingle" style="width:137px;max-width:137px"><div class="thumbimage" style="height:159px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Mysterium_Cosmographicum_solar_system_model.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Mysterium_Cosmographicum_solar_system_model.jpg/135px-Mysterium_Cosmographicum_solar_system_model.jpg" decoding="async" width="135" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Mysterium_Cosmographicum_solar_system_model.jpg/203px-Mysterium_Cosmographicum_solar_system_model.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Mysterium_Cosmographicum_solar_system_model.jpg/270px-Mysterium_Cosmographicum_solar_system_model.jpg 2x" data-file-width="3252" data-file-height="3832" /></a></span></div><div class="thumbcaption"><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler's</a> Platonic solid model of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a></div></div></div></div></div> <p>As mentioned above, the regular icosahedron is one of the five <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>. The regular polyhedra have been known since antiquity, but are named after <a href="/wiki/Plato" title="Plato">Plato</a> who, in his <a href="/wiki/Timaeus_(dialogue)" title="Timaeus (dialogue)"><i>Timaeus</i></a> dialogue, identified these with the five <i>elements</i>, whose elementary units were attributed these shapes: <a href="/wiki/Fire_(classical_element)" title="Fire (classical element)">fire</a> (tetrahedron), <a href="/wiki/Air_(classical_element)" title="Air (classical element)"> air</a> (octahedron), <a href="/wiki/Water_(classical_element)" title="Water (classical element)">water</a> (icosahedron), <a href="/wiki/Earth_(classical_element)" title="Earth (classical element)">earth</a> (cube) and the shape of the universe as a whole (dodecahedron). <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a> defined the Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length.<sup id="cite_ref-FOOTNOTEHeath1908262,_478,_480_33-0" class="reference"><a href="#cite_note-FOOTNOTEHeath1908262,_478,_480-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Following their identification with the elements by Plato, <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> in his <i><a href="/wiki/Harmonices_Mundi" title="Harmonices Mundi">Harmonices Mundi</a></i> sketched each of them, in particular, the regular icosahedron.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]_34-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> In his <i><a href="/wiki/Mysterium_Cosmographicum" title="Mysterium Cosmographicum">Mysterium Cosmographicum</a></i>, he also proposed a model of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> based on the placement of Platonic solids in a concentric sequence of increasing radius of the inscribed and circumscribed spheres whose radii gave the distance of the six known planets from the common center. The ordering of the solids, from innermost to outermost, consisted of: <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>, regular icosahedron, <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a>, <a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedron</a>, and <a href="/wiki/Cube" title="Cube">cube</a>.<sup id="cite_ref-FOOTNOTELivio2003147_35-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003147-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTESilvester2001[httpsbooksgooglecombooksidVtH_QG6scSUCpgPA141_140&ndash;141]Cundy1952-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESilvester2001[httpsbooksgooglecombooksidVtH_QG6scSUCpgPA141_140&ndash;141]Cundy1952_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSilvester2001">Silvester 2001</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VtH_QG6scSUC&pg=PA141">140–141</a>; <a href="#CITEREFCundy1952">Cundy 1952</a>.</span> </li> <li id="cite_note-FOOTNOTEBerman1971-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBerman1971_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBerman1971_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBerman1971">Berman 1971</a>.</span> </li> <li id="cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage70mode1upviewtheater_70]-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage70mode1upviewtheater_70]_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCromwell1997">Cromwell 1997</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/70/mode/1up?view=theater">70</a>.</span> </li> <li id="cite_note-FOOTNOTEShavinina2013[httpsbooksgooglecombooksidJcPd_JRc4FgCpgPA333_333]Cundy1952-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShavinina2013[httpsbooksgooglecombooksidJcPd_JRc4FgCpgPA333_333]Cundy1952_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShavinina2013">Shavinina 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JcPd_JRc4FgC&pg=PA333">333</a>; <a href="#CITEREFCundy1952">Cundy 1952</a>.</span> </li> <li id="cite_note-FOOTNOTEKappraff1991[httpsbooksgooglecombooksidtz76s0ZGFiQCpgPA475_475]-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKappraff1991[httpsbooksgooglecombooksidtz76s0ZGFiQCpgPA475_475]_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKappraff1991">Kappraff 1991</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tz76s0ZGFiQC&pg=PA475">475</a>.</span> </li> <li id="cite_note-FOOTNOTESteebHardyTanski2012[httpsbooksgooglecombooksidUdI7DQAAQBAJpgPA211_211]-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESteebHardyTanski2012[httpsbooksgooglecombooksidUdI7DQAAQBAJpgPA211_211]_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteebHardyTanski2012">Steeb, Hardy & Tanski 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UdI7DQAAQBAJ&pg=PA211">211</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFMacLean2007">MacLean 2007</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vINuAwAAQBAJ&pg=PA44">43–44</a>; <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, Table I(i), pp. 292–293. See column "<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 {}_{1}\!\mathrm {R} /\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{1}\!\mathrm {R} /\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064c055a3b71309163d3bd09f2c9c40a95a4d15a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.51ex; height:2.843ex;" alt="{\displaystyle {}_{1}\!\mathrm {R} /\ell }"></span>", where <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 {}_{1}\!\mathrm {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{1}\!\mathrm {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd61a947c8048d63ce10b32b59751713c4a54bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.378ex; height:2.509ex;" alt="{\displaystyle {}_{1}\!\mathrm {R} }"></span> is Coxeter's notation for the midradius, also noting that Coxeter uses <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 2\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a443fec94d4ffc53fd40d50d2c882927d08b0b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle 2\ell }"></span> as the edge length (see p. 2).</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFMacLean2007">MacLean 2007</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vINuAwAAQBAJ&pg=PA44">43–44</a></li><li><a href="#CITEREFBerman1971">Berman 1971</a></li></ul></div></span> </li> <li id="cite_note-FOOTNOTEHerz-Fischler2013[httpsbooksgooglecombooksidaYjXZJwLARQCpgPA138_138–140]-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHerz-Fischler2013[httpsbooksgooglecombooksidaYjXZJwLARQCpgPA138_138–140]_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerz-Fischler2013">Herz-Fischler 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aYjXZJwLARQC&pg=PA138">138–140</a>.</span> </li> <li id="cite_note-FOOTNOTESimmons2007[httpsbooksgooglecombooksid3KOst4Mon90CpgPA50_50]-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESimmons2007[httpsbooksgooglecombooksid3KOst4Mon90CpgPA50_50]_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSimmons2007">Simmons 2007</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3KOst4Mon90C&pg=PA50">50</a>.</span> </li> <li id="cite_note-FOOTNOTESutton2002[httpsbooksgooglecombooksidvgo7bTxDmIsCpgPA55_55]-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESutton2002[httpsbooksgooglecombooksidvgo7bTxDmIsCpgPA55_55]_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSutton2002">Sutton 2002</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55">55</a>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Numerical values for the volumes of the inscribed Platonic solids may be found in <a href="#CITEREFBukerEggleton1969">Buker & Eggleton 1969</a>.</span> </li> <li id="cite_note-FOOTNOTEJohnson1966See_table_II,_line_4.MacLean2007[httpsbooksgooglecombooksidvINuAwAAQBAJpgPA44_43&ndash;44]-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJohnson1966See_table_II,_line_4.MacLean2007[httpsbooksgooglecombooksidvINuAwAAQBAJpgPA44_43&ndash;44]_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJohnson1966">Johnson 1966</a>, See table II, line 4.; <a href="#CITEREFMacLean2007">MacLean 2007</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vINuAwAAQBAJ&pg=PA44">43–44</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFKlein1884">Klein 1884</a>. See <a href="/wiki/Icosahedral_symmetry#Related_geometries" title="Icosahedral symmetry">icosahedral symmetry: related geometries</a> for further history, and related symmetries on seven and eleven letters.</span> </li> <li id="cite_note-FOOTNOTEBickle2020[httpsbooksgooglecombooksid2sbVDwAAQBAJpgPA147_147]-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBickle2020[httpsbooksgooglecombooksid2sbVDwAAQBAJpgPA147_147]_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBickle2020">Bickle 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2sbVDwAAQBAJ&pg=PA147">147</a>.</span> </li> <li id="cite_note-FOOTNOTEHopkins2004-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHopkins2004_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHopkins2004">Hopkins 2004</a>.</span> </li> <li id="cite_note-FOOTNOTEHerrmannSally2013[httpsbooksgooglecombooksidb2fjR81h6yECpgPA257_257]-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHerrmannSally2013[httpsbooksgooglecombooksidb2fjR81h6yECpgPA257_257]_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHerrmannSally2013">Herrmann & Sally 2013</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b2fjR81h6yEC&pg=PA257">257</a>.</span> </li> <li id="cite_note-FOOTNOTECoxeterdu_ValFlatherPetrie19384-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeterdu_ValFlatherPetrie19384_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeterdu_ValFlatherPetrie1938">Coxeter et al. 1938</a>, p. 4.</span> </li> <li id="cite_note-FOOTNOTEBorovik20068–9§5._How_to_draw_an_icosahedron_on_a_blackboard-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBorovik20068–9§5._How_to_draw_an_icosahedron_on_a_blackboard_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBorovik2006">Borovik 2006</a>, pp. 8–9, §5. How to draw an icosahedron on a blackboard.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Reciprocally, the edge length of a cube inscribed in a dodecahedron is in the golden ratio to the dodecahedron's edge length. The cube's edges lie in pentagonal face planes of the dodecahedron as <a href="/wiki/Pentagon#Regular_pentagons" title="Pentagon">regular pentagon diagonals</a>, which are always in the golden ratio to the regular pentagon's edge. When a cube is inscribed in a dodecahedron and an icosahedron is inscribed in the cube, the dodecahedron and icosahedron that do not share any vertices have the same edge length.</span> </li> <li id="cite_note-FOOTNOTECoxeterdu_ValFlatherPetrie19388&ndash;26-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeterdu_ValFlatherPetrie19388&ndash;26_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeterdu_ValFlatherPetrie1938">Coxeter et al. 1938</a>, p. 8–26.</span> </li> <li id="cite_note-FOOTNOTESmith1958[httpsbooksgooglecombooksiduTytJGnTf1kCpgPA295_295]Minas-Nerpel2007-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESmith1958[httpsbooksgooglecombooksiduTytJGnTf1kCpgPA295_295]Minas-Nerpel2007_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSmith1958">Smith 1958</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTytJGnTf1kC&pg=PA295">295</a>; <a href="#CITEREFMinas-Nerpel2007">Minas-Nerpel 2007</a>.</span> </li> <li id="cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage4mode1upviewtheater_4]-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage4mode1upviewtheater_4]_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCromwell1997">Cromwell 1997</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/4/mode/1up?view=theater">4</a>.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.gmdice.com/pages/dungeons-dragons-dice">"Dungeons & Dragons Dice"</a>. <i>gmdice.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">August 20,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=gmdice.com&rft.atitle=Dungeons+%26+Dragons+Dice&rft_id=https%3A%2F%2Fwww.gmdice.com%2Fpages%2Fdungeons-dragons-dice&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEFlanaganGregory2015[httpsbooksgooglecombooksidhViuEAAAQBAJpgPA85_85]-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFlanaganGregory2015[httpsbooksgooglecombooksidhViuEAAAQBAJpgPA85_85]_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFlanaganGregory2015">Flanagan & Gregory 2015</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hViuEAAAQBAJ&pg=PA85">85</a>.</span> </li> <li id="cite_note-FOOTNOTEStraussStrauss200835&ndash;62-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStraussStrauss200835&ndash;62_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStraussStrauss2008">Strauss & Strauss 2008</a>, p. 35–62.</span> </li> <li id="cite_note-FOOTNOTEHaeckel1904Cromwell1997[httpsarchiveorgdetailspolyhedra0000crompage6mode1upviewtheater_6]-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHaeckel1904Cromwell1997[httpsarchiveorgdetailspolyhedra0000crompage6mode1upviewtheater_6]_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHaeckel1904">Haeckel 1904</a>; <a href="#CITEREFCromwell1997">Cromwell 1997</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/6/mode/1up?view=theater">6</a>.</span> </li> <li id="cite_note-FOOTNOTESpokoyny2013-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESpokoyny2013_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSpokoyny2013">Spokoyny 2013</a>.</span> </li> <li id="cite_note-FOOTNOTEHofmeister2004-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHofmeister2004_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHofmeister2004">Hofmeister 2004</a>.</span> </li> <li id="cite_note-FOOTNOTEDronskowskiKikkawaStein2017[httpsbooksgooglecombookside0VBDwAAQBAJpgPA436_435&ndash;436]-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDronskowskiKikkawaStein2017[httpsbooksgooglecombookside0VBDwAAQBAJpgPA436_435&ndash;436]_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDronskowskiKikkawaStein2017">Dronskowski, Kikkawa & Stein 2017</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e0VBDwAAQBAJ&pg=PA436">435–436</a>.</span> </li> <li id="cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage7mode1upviewtheater_7]-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage7mode1upviewtheater_7]_31-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCromwell1997">Cromwell 1997</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/7/mode/1up?view=theater">7</a>.</span> </li> <li id="cite_note-FOOTNOTEWhyte1952-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWhyte1952_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWhyte1952">Whyte 1952</a>.</span> </li> <li id="cite_note-FOOTNOTEHeath1908262,_478,_480-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHeath1908262,_478,_480_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeath1908">Heath 1908</a>, p. 262, 478, 480.</span> </li> <li id="cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCromwell1997">Cromwell 1997</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/55">55</a>.</span> </li> <li id="cite_note-FOOTNOTELivio2003147-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELivio2003147_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLivio2003">Livio 2003</a>, p. 147.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerman1971" class="citation journal cs1">Berman, Martin (1971). "Regular-faced convex polyhedra". <i>Journal of the Franklin Institute</i>. <b>291</b> (5): 329–352. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2871%2990071-8">10.1016/0016-0032(71)90071-8</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0290245">0290245</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Franklin+Institute&rft.atitle=Regular-faced+convex+polyhedra&rft.volume=291&rft.issue=5&rft.pages=329-352&rft.date=1971&rft_id=info%3Adoi%2F10.1016%2F0016-0032%2871%2990071-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D290245%23id-name%3DMR&rft.aulast=Berman&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBickle2020" class="citation book cs1">Bickle, Allan (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2sbVDwAAQBAJ"><i>Fundamentals of Graph Theory</i></a>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781470455491" title="Special:BookSources/9781470455491"><bdi>9781470455491</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Graph+Theory&rft.pub=American+Mathematical+Society&rft.date=2020&rft.isbn=9781470455491&rft.aulast=Bickle&rft.aufirst=Allan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2sbVDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorovik2006" class="citation book cs1"><a href="/wiki/Alexandre_Borovik" title="Alexandre Borovik">Borovik, Alexandre</a> (2006). "Coxeter Theory: The Cognitive Aspects". In Davis, Chandler; Ellers, Erich (eds.). <a rel="nofollow" class="external text" href="https://www.academia.edu/26091464"><i>The Coxeter Legacy</i></a>. Providence, Rhode Island: American Mathematical Society. pp. 17–43. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0821837221" title="Special:BookSources/978-0821837221"><bdi>978-0821837221</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Coxeter+Theory%3A+The+Cognitive+Aspects&rft.btitle=The+Coxeter+Legacy&rft.place=Providence%2C+Rhode+Island&rft.pages=17-43&rft.pub=American+Mathematical+Society&rft.date=2006&rft.isbn=978-0821837221&rft.aulast=Borovik&rft.aufirst=Alexandre&rft_id=https%3A%2F%2Fwww.academia.edu%2F26091464&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBukerEggleton1969" class="citation journal cs1">Buker, W. E.; Eggleton, R. B. (1969). "The Platonic Solids (Solution to problem E2053)". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>76</b> (2): 192. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2317282">10.2307/2317282</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2317282">2317282</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Platonic+Solids+%28Solution+to+problem+E2053%29&rft.volume=76&rft.issue=2&rft.pages=192&rft.date=1969&rft_id=info%3Adoi%2F10.2307%2F2317282&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2317282%23id-name%3DJSTOR&rft.aulast=Buker&rft.aufirst=W.+E.&rft.au=Eggleton%2C+R.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1973" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1973). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16">"2.1 Regular polyhedra; 2.2 Reciprocation"</a>. <i><a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)">Regular Polytopes</a></i> (3rd ed.). Dover Publications. pp. 16–17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61480-8" title="Special:BookSources/0-486-61480-8"><bdi>0-486-61480-8</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0370327">0370327</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.1+Regular+polyhedra%3B+2.2+Reciprocation&rft.btitle=Regular+Polytopes&rft.pages=16-17&rft.edition=3rd&rft.pub=Dover+Publications&rft.date=1973&rft.isbn=0-486-61480-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0370327%23id-name%3DMR&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiWvXsVInpgMC%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeterdu_ValFlatherPetrie1938" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a>; <a href="/wiki/Patrick_du_Val" title="Patrick du Val">du Val, Patrick</a>; Flather, H. T.; <a href="/wiki/John_Flinders_Petrie" title="John Flinders Petrie">Petrie, J. F.</a> (1938). <i><a href="/wiki/The_Fifty-Nine_Icosahedra" title="The Fifty-Nine Icosahedra">The Fifty-Nine Icosahedra</a></i>. Vol. 6. University of Toronto Studies (Mathematical Series).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fifty-Nine+Icosahedra&rft.pub=University+of+Toronto+Studies+%28Mathematical+Series%29&rft.date=1938&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rft.au=du+Val%2C+Patrick&rft.au=Flather%2C+H.+T.&rft.au=Petrie%2C+J.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCromwell1997" class="citation book cs1">Cromwell, Peter R. (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom"><i>Polyhedra</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55432-9" title="Special:BookSources/978-0-521-55432-9"><bdi>978-0-521-55432-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyhedra&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-55432-9&rft.aulast=Cromwell&rft.aufirst=Peter+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpolyhedra0000crom&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCundy1952" class="citation journal cs1">Cundy, H. Martyn (1952). "Deltahedra". <i>The Mathematical Gazette</i>. <b>36</b> (318): 263–266. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3608204">10.2307/3608204</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3608204">3608204</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250435684">250435684</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Deltahedra&rft.volume=36&rft.issue=318&rft.pages=263-266&rft.date=1952&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250435684%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3608204%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3608204&rft.aulast=Cundy&rft.aufirst=H.+Martyn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDronskowskiKikkawaStein2017" class="citation book cs1">Dronskowski, Richard; Kikkawa, Shinichi; Stein, Andreas (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e0VBDwAAQBAJ"><i>Handbook of Solid State Chemistry, 6 Volume Set</i></a>. John Sons & Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-69103-6" title="Special:BookSources/978-3-527-69103-6"><bdi>978-3-527-69103-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Solid+State+Chemistry%2C+6+Volume+Set&rft.pub=John+Sons+%26+Wiley&rft.date=2017&rft.isbn=978-3-527-69103-6&rft.aulast=Dronskowski&rft.aufirst=Richard&rft.au=Kikkawa%2C+Shinichi&rft.au=Stein%2C+Andreas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De0VBDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlanaganGregory2015" class="citation book cs1">Flanagan, Kieran; Gregory, Dan (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hViuEAAAQBAJ"><i>Selfish, Scared and Stupid: Stop Fighting Human Nature and Increase Your Performance, Engagement and Influence</i></a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780730312796" title="Special:BookSources/9780730312796"><bdi>9780730312796</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Selfish%2C+Scared+and+Stupid%3A+Stop+Fighting+Human+Nature+and+Increase+Your+Performance%2C+Engagement+and+Influence&rft.pub=John+Wiley+%26+Sons&rft.date=2015&rft.isbn=9780730312796&rft.aulast=Flanagan&rft.aufirst=Kieran&rft.au=Gregory%2C+Dan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhViuEAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaeckel1904" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Ernst_Haeckel" title="Ernst Haeckel">Haeckel, E.</a> (1904). <i><a href="/wiki/Kunstformen_der_Natur" title="Kunstformen der Natur">Kunstformen der Natur</a></i> (in German).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kunstformen+der+Natur&rft.date=1904&rft.aulast=Haeckel&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span> See <a rel="nofollow" class="external text" href="http://www.biolib.de/haeckel/kunstformen/index.html">here</a> for an online book.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerrmannSally2013" class="citation book cs1">Herrmann, Diane L.; Sally, Paul J. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b2fjR81h6yEC"><i>Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-5464-1" title="Special:BookSources/978-1-4665-5464-1"><bdi>978-1-4665-5464-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number%2C+Shape%2C+%26+Symmetry%3A+An+Introduction+to+Number+Theory%2C+Geometry%2C+and+Group+Theory&rft.pub=CRC+Press&rft.date=2013&rft.isbn=978-1-4665-5464-1&rft.aulast=Herrmann&rft.aufirst=Diane+L.&rft.au=Sally%2C+Paul+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Db2fjR81h6yEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1908" class="citation book cs1"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Thomas L.</a> (1908). <a rel="nofollow" class="external text" href="https://archive.org/details/thirteenbookseu03heibgoog/page/480"><i>The Thirteen Books of Euclid's Elements</i></a> (3rd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+Euclid%27s+Elements&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=1908&rft.aulast=Heath&rft.aufirst=Thomas+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthirteenbookseu03heibgoog%2Fpage%2F480&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerz-Fischler2013" class="citation book cs1">Herz-Fischler, Roger (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aYjXZJwLARQC"><i>A Mathematical History of the Golden Number</i></a>. Courier Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486152325" title="Special:BookSources/9780486152325"><bdi>9780486152325</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+History+of+the+Golden+Number&rft.pub=Courier+Dover+Publications&rft.date=2013&rft.isbn=9780486152325&rft.aulast=Herz-Fischler&rft.aufirst=Roger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaYjXZJwLARQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofmeister2004" class="citation journal cs1">Hofmeister, H. (2004). "Fivefold Twinned Nanoparticles". <i>Encyclopedia of Nanoscience and Nanotechnology</i>. <b>3</b>: 431–452.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Encyclopedia+of+Nanoscience+and+Nanotechnology&rft.atitle=Fivefold+Twinned+Nanoparticles&rft.volume=3&rft.pages=431-452&rft.date=2004&rft.aulast=Hofmeister&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHopkins2004" class="citation journal cs1">Hopkins, Brian (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FS0161171204307118">"Hamiltonian paths on Platonic graphs"</a>. <i><a href="/wiki/International_Journal_of_Mathematics_and_Mathematical_Sciences" title="International Journal of Mathematics and Mathematical Sciences">International Journal of Mathematics and Mathematical Sciences</a></i>. <b>2004</b> (30): 1613–1616. <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/10.1155%2FS0161171204307118">10.1155/S0161171204307118</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Mathematics+and+Mathematical+Sciences&rft.atitle=Hamiltonian+paths+on+Platonic+graphs&rft.volume=2004&rft.issue=30&rft.pages=1613-1616&rft.date=2004&rft_id=info%3Adoi%2F10.1155%2FS0161171204307118&rft.aulast=Hopkins&rft.aufirst=Brian&rft_id=https%3A%2F%2Fdoi.org%2F10.1155%252FS0161171204307118&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson1966" class="citation journal cs1"><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">Johnson, Norman W.</a> (1966). 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Roach, Peter; Hartmann, James; Setter, Jane (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%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKappraff1991" class="citation book cs1">Kappraff, Jay (1991). <i>Connections: The Geometric Bridge Between Art and Science</i> (2nd ed.). <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-281-139-4" title="Special:BookSources/978-981-281-139-4"><bdi>978-981-281-139-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Connections%3A+The+Geometric+Bridge+Between+Art+and+Science&rft.edition=2nd&rft.pub=World+Scientific&rft.date=1991&rft.isbn=978-981-281-139-4&rft.aulast=Kappraff&rft.aufirst=Jay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></ref></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1888" class="citation book cs1"><a href="/wiki/Felix_Klein" title="Felix Klein">Klein, Felix</a> (1888). <a rel="nofollow" class="external text" href="http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03070001"><i>Lectures on the ikosahedron and the solution of equations of the fifth degree</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49528-6" title="Special:BookSources/978-0-486-49528-6"><bdi>978-0-486-49528-6</bdi></a>, Dover edition</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+the+ikosahedron+and+the+solution+of+equations+of+the+fifth+degree&rft.pub=Courier+Corporation&rft.date=1888&rft.isbn=978-0-486-49528-6&rft.aulast=Klein&rft.aufirst=Felix&rft_id=http%3A%2F%2Fdigital.library.cornell.edu%2Fcgi%2Ft%2Ftext%2Ftext-idx%3Fc%3Dmath%3Bcc%3Dmath%3Bview%3Dtoc%3Bsubview%3Dshort%3Bidno%3D03070001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: postscript (<a href="/wiki/Category:CS1_maint:_postscript" title="Category:CS1 maint: postscript">link</a>)</span>, translated from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1884" class="citation book cs1">Klein, Felix (1884). <a rel="nofollow" class="external text" href="https://archive.org/details/vorlesungenberd00kleigoog"><i>Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade</i></a>. Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+das+Ikosaeder+und+die+Aufl%C3%B6sung+der+Gleichungen+vom+f%C3%BCnften+Grade&rft.pub=Teubner&rft.date=1884&rft.aulast=Klein&rft.aufirst=Felix&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fvorlesungenberd00kleigoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLivio2003" class="citation book cs1"><a href="/wiki/Mario_Livio" title="Mario Livio">Livio, Mario</a> (2003) [2002]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bUARfgWRH14C"><i>The Golden Ratio: The Story of Phi, the World's Most Astonishing Number</i></a> (1st trade paperback ed.). New York City: <a href="/wiki/Random_House" title="Random House">Broadway Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7679-0816-0" title="Special:BookSources/978-0-7679-0816-0"><bdi>978-0-7679-0816-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Golden+Ratio%3A+The+Story+of+Phi%2C+the+World%27s+Most+Astonishing+Number&rft.place=New+York+City&rft.edition=1st+trade+paperback&rft.pub=Broadway+Books&rft.date=2003&rft.isbn=978-0-7679-0816-0&rft.aulast=Livio&rft.aufirst=Mario&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbUARfgWRH14C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacLean2007" class="citation book cs1">MacLean, Kenneth J. M. 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Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780883855614" title="Special:BookSources/9780883855614"><bdi>9780883855614</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+Gems%3A+Brief+Lives+and+Memorable+Mathematics&rft.pub=Mathematical+Association+of+America&rft.date=2007&rft.isbn=9780883855614&rft.aulast=Simmons&rft.aufirst=George+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1958" class="citation book cs1">Smith, David E. (1958). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uTytJGnTf1kC"><i>History of Mathematics</i></a>. Vol. 2. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-20430-8" title="Special:BookSources/0-486-20430-8"><bdi>0-486-20430-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.pub=Dover+Publications&rft.date=1958&rft.isbn=0-486-20430-8&rft.aulast=Smith&rft.aufirst=David+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuTytJGnTf1kC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpokoyny2013" class="citation journal cs1"><a href="/wiki/Alexander_M._Spokoyny" title="Alexander M. Spokoyny">Spokoyny, A. M.</a> (2013). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3845684">"New Ligand Platforms Featuring Boron-Rich Clusters as Organomimetic Sbstituents"</a>. <i>Pure and Applied Chemistry</i>. <b>85</b> (5): 903–919. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2FPAC-CON-13-01-13">10.1351/PAC-CON-13-01-13</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3845684">3845684</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24311823">24311823</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pure+and+Applied+Chemistry&rft.atitle=New+Ligand+Platforms+Featuring+Boron-Rich+Clusters+as+Organomimetic+Sbstituents&rft.volume=85&rft.issue=5&rft.pages=903-919&rft.date=2013&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3845684%23id-name%3DPMC&rft_id=info%3Apmid%2F24311823&rft_id=info%3Adoi%2F10.1351%2FPAC-CON-13-01-13&rft.aulast=Spokoyny&rft.aufirst=A.+M.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3845684&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinmitzManchester2011" class="citation book cs1">Steinmitz, Nicole F.; Manchester, Marianne (2011). <i>Viral Nanoparticles: Tools for Material Science and Biomedicine</i>. Pan Stanford Publisher. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4267-94-6" title="Special:BookSources/978-981-4267-94-6"><bdi>978-981-4267-94-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Viral+Nanoparticles%3A+Tools+for+Material+Science+and+Biomedicine&rft.pub=Pan+Stanford+Publisher&rft.date=2011&rft.isbn=978-981-4267-94-6&rft.aulast=Steinmitz&rft.aufirst=Nicole+F.&rft.au=Manchester%2C+Marianne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStraussStrauss2008" class="citation book cs1">Strauss, James H.; Strauss, Ellen G. (2008). "The Structure of Viruses". <i>Viruses and Human Disease</i>. 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Wooden Books. Bloomsbury Publishing USA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780802713865" title="Special:BookSources/9780802713865"><bdi>9780802713865</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Platonic+%26+Archimedean+Solids&rft.series=Wooden+Books&rft.pub=Bloomsbury+Publishing+USA&rft.date=2002&rft.isbn=9780802713865&rft.aulast=Sutton&rft.aufirst=Daud&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dvgo7bTxDmIsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteebHardyTanski2012" class="citation book cs1">Steeb, Willi-hans; Hardy, Yorick; Tanski, Igor (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UdI7DQAAQBAJ"><i>Problems And Solutions For Groups, Lie Groups, Lie Algebras With Applications</i></a>. World Scientific Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789813104112" title="Special:BookSources/9789813104112"><bdi>9789813104112</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Problems+And+Solutions+For+Groups%2C+Lie+Groups%2C+Lie+Algebras+With+Applications&rft.pub=World+Scientific+Publishing+Company&rft.date=2012&rft.isbn=9789813104112&rft.aulast=Steeb&rft.aufirst=Willi-hans&rft.au=Hardy%2C+Yorick&rft.au=Tanski%2C+Igor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUdI7DQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhyte1952" class="citation journal cs1">Whyte, L. L. (1952). "Unique arrangements of points on a sphere". <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>59</b> (9): 606–611. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1952.11988207">10.1080/00029890.1952.11988207</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2306764">2306764</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0050303">0050303</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Unique+arrangements+of+points+on+a+sphere&rft.volume=59&rft.issue=9&rft.pages=606-611&rft.date=1952&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D50303%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2306764%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F00029890.1952.11988207&rft.aulast=Whyte&rft.aufirst=L.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_icosahedron&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Icosahedron" class="extiw" title="commons:Category:Icosahedron">Icosahedron</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 <i>Encyclopædia Britannica</i></a> article "<span style="font-weight:bold;"><a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Icosahedron" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Icosahedron">Icosahedron</a></span>".</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/icosahedron" class="extiw" title="wiktionary:icosahedron">icosahedron</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlitzing" class="citation web cs1">Klitzing, Richard. <a rel="nofollow" class="external text" href="https://bendwavy.org/klitzing/dimensions/polyhedra.htm">"3D convex uniform polyhedra x3o5o – ike"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=3D+convex+uniform+polyhedra+x3o5o+%E2%80%93+ike&rft.aulast=Klitzing&rft.aufirst=Richard&rft_id=https%3A%2F%2Fbendwavy.org%2Fklitzing%2Fdimensions%2Fpolyhedra.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartley" class="citation web cs1">Hartley, Michael. <a rel="nofollow" class="external text" href="http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=PKYcLAHcKk9lKCGWdoeJHvnDe7jSJyKXIghPYlKcV5PxkeWqGqwmeoyXUilmvzkXsQnxEoduPHUYFqOf20B8EwLz8CufLOUuc4N5&name=Icosahedron#applet">"Dr Mike's Math Games for Kids"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dr+Mike%27s+Math+Games+for+Kids&rft.aulast=Hartley&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.dr-mikes-math-games-for-kids.com%2Fpolyhedral-nets.html%3Fnet%3DPKYcLAHcKk9lKCGWdoeJHvnDe7jSJyKXIghPYlKcV5PxkeWqGqwmeoyXUilmvzkXsQnxEoduPHUYFqOf20B8EwLz8CufLOUuc4N5%26name%3DIcosahedron%23applet&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+icosahedron" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.kjmaclean.com/Geometry/GeometryHome.html">K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra</a></li> <li><a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> The Encyclopedia of Polyhedra</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040922084928/http://www.tulane.edu/~dmsander/WWW/335/335Structure.html">Tulane.edu</a> A discussion of viral structure and the icosahedron</li> <li><a rel="nofollow" class="external text" href="https://www.flickr.com/photos/pascalin/sets/72157594234292561/">Origami Polyhedra</a> – Models made with Modular Origami</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=r6JQWCi_lNM">Video of icosahedral mirror sculpture</a></li> <li><a rel="nofollow" class="external autonumber" href="https://archive.today/20121224154747/http://web.uct.ac.za/depts/mmi/stannard/virarch.html">[1]</a> Principle of virus 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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 href="/wiki/Icosahedron" title="Icosahedron">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 class="mw-selflink selflink">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> <table class="wikitable mw-collapsible"> <tbody><tr> <th colspan="13" style="background:lightsteelblue;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-collapse navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Polytopes" title="Template:Polytopes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polytopes" title="Template talk:Polytopes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polytopes" title="Special:EditPage/Template:Polytopes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div class="navbar-ct-mini">Fundamental convex <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a> and <a href="/wiki/Uniform_polytope" title="Uniform polytope">uniform polytopes</a> in dimensions 2–10</div> </th></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Coxeter_group#Finite_Coxeter_groups" title="Coxeter group">Family</a> </th> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group"><i>A</i><sub><i>n</i></sub></a> </td> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group"><i>B</i><sub><i>n</i></sub></a> </td> <td style="background:gainsboro;"><span style="background-color: #f0f0e0; color:;"><i>I</i><sub>2</sub>(p)</span> / <a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group"><i>D</i><sub><i>n</i></sub></a> </td> <td style="background:gainsboro;"><span style="background-color: #f0e0e0; color:;"><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)"><i>E</i><sub>6</sub></a> / <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)"><i>E</i><sub>7</sub></a> / <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)"><i>E</i><sub>8</sub></a></span> / <span style="background-color: #e0f0e0; color:;"><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)"><i>F</i><sub>4</sub></a></span> / <span style="background-color: #e0e0f0; color:;"><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)"><i>G</i><sub>2</sub></a></span> </td> <td style="background:gainsboro;"><a href="/wiki/H4_(mathematics)" class="mw-redirect" title="H4 (mathematics)">H<sub>n</sub></a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a> </th> <td><a href="/wiki/Equilateral_triangle" title="Equilateral triangle">Triangle</a> </td> <td><a href="/wiki/Square" title="Square">Square</a> </td> <td style="background:#f0f0e0;"><a href="/wiki/Regular_polygon" title="Regular polygon">p-gon</a> </td> <td style="background:#e0e0f0;"><a href="/wiki/Hexagon" title="Hexagon">Hexagon</a> </td> <td><a href="/wiki/Pentagon" title="Pentagon">Pentagon</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">Uniform polyhedron</a> </th> <td style="background:whitesmoke;"><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a> </td> <td style="background:whitesmoke;"><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a> • <a href="/wiki/Cube" title="Cube">Cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/Tetrahedron" title="Tetrahedron">Demicube</a> </td> <td style="background:whitesmoke;"> </td> <td style="background:whitesmoke;"><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Dodecahedron</a> • <a class="mw-selflink selflink">Icosahedron</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polychoron" class="mw-redirect" title="Uniform polychoron">Uniform polychoron</a> </th> <td><a href="/wiki/5-cell" title="5-cell">Pentachoron</a> </td> <td><a href="/wiki/16-cell" title="16-cell">16-cell</a> • <a href="/wiki/Tesseract" title="Tesseract">Tesseract</a> </td> <td><a href="/wiki/16-cell" title="16-cell">Demitesseract</a> </td> <td style="background:#e0f0e0;"><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td> <td><a href="/wiki/120-cell" title="120-cell">120-cell</a> • <a href="/wiki/600-cell" title="600-cell">600-cell</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_5-polytope" title="Uniform 5-polytope">Uniform 5-polytope</a> </th> <td style="background:whitesmoke;"><a href="/wiki/5-simplex" title="5-simplex">5-simplex</a> </td> <td style="background:whitesmoke;"><a href="/wiki/5-orthoplex" title="5-orthoplex">5-orthoplex</a> • <a href="/wiki/5-cube" title="5-cube">5-cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/5-demicube" title="5-demicube">5-demicube</a> </td> <td style="background:whitesmoke;"> </td> <td style="background:whitesmoke;"> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_6-polytope" title="Uniform 6-polytope">Uniform 6-polytope</a> </th> <td><a href="/wiki/6-simplex" title="6-simplex">6-simplex</a> </td> <td><a href="/wiki/6-orthoplex" title="6-orthoplex">6-orthoplex</a> • <a href="/wiki/6-cube" title="6-cube">6-cube</a> </td> <td><a href="/wiki/6-demicube" title="6-demicube">6-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_22_polytope" title="1 22 polytope">1<sub>22</sub></a> • <a href="/wiki/2_21_polytope" title="2 21 polytope">2<sub>21</sub></a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_7-polytope" title="Uniform 7-polytope">Uniform 7-polytope</a> </th> <td style="background:whitesmoke;"><a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> </td> <td style="background:whitesmoke;"><a href="/wiki/7-orthoplex" title="7-orthoplex">7-orthoplex</a> • <a href="/wiki/7-cube" title="7-cube">7-cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/7-demicube" title="7-demicube">7-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_32_polytope" title="1 32 polytope">1<sub>32</sub></a> • <a href="/wiki/2_31_polytope" title="2 31 polytope">2<sub>31</sub></a> • <a href="/wiki/3_21_polytope" title="3 21 polytope">3<sub>21</sub></a> </td> <td style="background:whitesmoke;"> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_8-polytope" title="Uniform 8-polytope">Uniform 8-polytope</a> </th> <td><a href="/wiki/8-simplex" title="8-simplex">8-simplex</a> </td> <td><a href="/wiki/8-orthoplex" title="8-orthoplex">8-orthoplex</a> • <a href="/wiki/8-cube" title="8-cube">8-cube</a> </td> <td><a href="/wiki/8-demicube" title="8-demicube">8-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_42_polytope" title="1 42 polytope">1<sub>42</sub></a> • <a href="/wiki/2_41_polytope" title="2 41 polytope">2<sub>41</sub></a> • <a href="/wiki/4_21_polytope" title="4 21 polytope">4<sub>21</sub></a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_9-polytope" title="Uniform 9-polytope">Uniform 9-polytope</a> </th> <td style="background:whitesmoke;"><a href="/wiki/9-simplex" title="9-simplex">9-simplex</a> </td> <td style="background:whitesmoke;"><a href="/wiki/9-orthoplex" title="9-orthoplex">9-orthoplex</a> • <a href="/wiki/9-cube" title="9-cube">9-cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/9-demicube" title="9-demicube">9-demicube</a> </td> <td style="background:whitesmoke;"> </td> <td style="background:whitesmoke;"> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_10-polytope" title="Uniform 10-polytope">Uniform 10-polytope</a> </th> <td><a href="/wiki/10-simplex" title="10-simplex">10-simplex</a> </td> <td><a href="/wiki/10-orthoplex" title="10-orthoplex">10-orthoplex</a> • <a href="/wiki/10-cube" title="10-cube">10-cube</a> </td> <td><a href="/wiki/10-demicube" title="10-demicube">10-demicube</a> </td> <td> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;">Uniform <i>n</i>-<a href="/wiki/Polytope" title="Polytope">polytope</a> </th> <td style="background:whitesmoke;"><i>n</i>-<a href="/wiki/Simplex" title="Simplex">simplex</a> </td> <td style="background:whitesmoke;"><i>n</i>-<a href="/wiki/Cross-polytope" title="Cross-polytope">orthoplex</a> • <i>n</i>-<a href="/wiki/Hypercube" title="Hypercube">cube</a> </td> <td style="background:whitesmoke;"><i>n</i>-<a href="/wiki/Demihypercube" title="Demihypercube">demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/Uniform_1_k2_polytope" title="Uniform 1 k2 polytope">1<sub>k2</sub></a> • <a href="/wiki/Uniform_2_k1_polytope" title="Uniform 2 k1 polytope">2<sub>k1</sub></a> • <a href="/wiki/Uniform_k_21_polytope" title="Uniform k 21 polytope">k<sub>21</sub></a> </td> <td style="background:whitesmoke;"><i>n</i>-<a href="/wiki/Pentagonal_polytope" title="Pentagonal polytope">pentagonal polytope</a> </td></tr> <tr style="text-align:center;"> <th colspan="13" style="background:gainsboro;">Topics: <a href="/wiki/Polytope_families" title="Polytope families">Polytope families</a> • <a href="/wiki/Regular_polytope" title="Regular polytope">Regular polytope</a> • <a href="/wiki/List_of_regular_polytopes_and_compounds" class="mw-redirect" title="List of regular polytopes and compounds">List of regular polytopes and compounds</a> </th></tr></tbody></table> <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 class="mw-selflink selflink">(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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Regular_icosahedron&oldid=1258884486">https://en.wikipedia.org/w/index.php?title=Regular_icosahedron&oldid=1258884486</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div 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alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-2jqt7","wgBackendResponseTime":156,"wgPageParseReport":{"limitreport":{"cputime":"0.981","walltime":"1.238","ppvisitednodes":{"value":4869,"limit":1000000},"postexpandincludesize":{"value":130915,"limit":2097152},"templateargumentsize":{"value":5796,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":11,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":142729,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 877.949 1 -total"," 16.51% 144.940 1 Template:Reflist"," 14.32% 125.742 25 Template:Cite_book"," 12.91% 113.314 3 Template:Navbox"," 11.77% 103.320 3 Template:Cite_web"," 11.12% 97.634 1 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