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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Regular_dodecahedron" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Regular_dodecahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Regular dodecahedron</span> </div> </a> <ul id="toc-Regular_dodecahedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_pentagonal_dodecahedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_pentagonal_dodecahedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Other pentagonal dodecahedra</span> </div> </a> <button aria-controls="toc-Other_pentagonal_dodecahedra-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other pentagonal dodecahedra subsection</span> </button> <ul id="toc-Other_pentagonal_dodecahedra-sublist" class="vector-toc-list"> <li id="toc-Pyritohedron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pyritohedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Pyritohedron</span> </div> </a> <ul id="toc-Pyritohedron-sublist" class="vector-toc-list"> <li id="toc-Crystal_pyrite" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Crystal_pyrite"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Crystal pyrite</span> </div> </a> <ul id="toc-Crystal_pyrite-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartesian_coordinates" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cartesian_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Cartesian coordinates</span> </div> </a> <ul id="toc-Cartesian_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_freedom" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Geometric_freedom"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Geometric freedom</span> </div> </a> <ul id="toc-Geometric_freedom-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tetartoid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tetartoid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Tetartoid</span> </div> </a> <ul id="toc-Tetartoid-sublist" class="vector-toc-list"> <li id="toc-Cartesian_coordinates_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cartesian_coordinates_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Cartesian coordinates</span> </div> </a> <ul id="toc-Cartesian_coordinates_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_freedom_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Geometric_freedom_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Geometric freedom</span> </div> </a> <ul id="toc-Geometric_freedom_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dual_of_triangular_gyrobianticupola" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dual_of_triangular_gyrobianticupola"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Dual of triangular gyrobianticupola</span> </div> </a> <ul id="toc-Dual_of_triangular_gyrobianticupola-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rhombic_dodecahedron" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rhombic_dodecahedron"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rhombic dodecahedron</span> </div> </a> <ul id="toc-Rhombic_dodecahedron-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_dodecahedra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_dodecahedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other dodecahedra</span> </div> </a> <ul id="toc-Other_dodecahedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Practical_usage" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Practical_usage"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Practical usage</span> </div> </a> <ul id="toc-Practical_usage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">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">Dodecahedron</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 55 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-55" 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">55 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Dodeka%C3%ABder" title="Dodekaëder – Afrikaans" lang="af" hreflang="af" data-title="Dodekaëder" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%AB%D9%86%D8%A7_%D8%B9%D8%B4%D8%B1%D9%8A_%D8%B3%D8%B7%D9%88%D8%AD" title="اثنا عشري سطوح – Arabic" lang="ar" hreflang="ar" data-title="اثنا عشري سطوح" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Dodekaedr" title="Dodekaedr – Azerbaijani" lang="az" hreflang="az" data-title="Dodekaedr" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%94%D0%BE%D0%B4%D0%B5%D0%BA%D0%B0%D1%8D%D0%B4%D1%80" title="Додекаэдр – Bashkir" lang="ba" hreflang="ba" data-title="Додекаэдр" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B0%D0%B4%D1%8D%D0%BA%D0%B0%D1%8D%D0%B4%D1%80" title="Дадэкаэдр – Belarusian" lang="be" hreflang="be" data-title="Дадэкаэдр" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B2%D0%B0%D0%BD%D0%B0%D0%B4%D0%B5%D1%81%D0%B5%D1%82%D0%BE%D1%81%D1%82%D0%B5%D0%BD" title="Дванадесетостен – Bulgarian" lang="bg" hreflang="bg" data-title="Дванадесетостен" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dodec%C3%A0edre" title="Dodecàedre – Catalan" lang="ca" hreflang="ca" data-title="Dodecàedre" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%BE%D0%B4%D0%B5%D0%BA%D0%B0%D1%8D%D0%B4%D1%80" title="Додекаэдр – Chuvash" lang="cv" hreflang="cv" data-title="Додекаэдр" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dvan%C3%A1ctist%C4%9Bn" title="Dvanáctistěn – Czech" lang="cs" hreflang="cs" data-title="Dvanáctistěn" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dodecahedron" title="Dodecahedron – Welsh" lang="cy" hreflang="cy" data-title="Dodecahedron" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Dodekaeder" title="Dodekaeder – Danish" lang="da" hreflang="da" data-title="Dodekaeder" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dodekaeder" title="Dodekaeder – German" lang="de" hreflang="de" data-title="Dodekaeder" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%89%CE%B4%CE%B5%CE%BA%CE%AC%CE%B5%CE%B4%CF%81%CE%BF" title="Δωδεκάεδρο – Greek" lang="el" hreflang="el" data-title="Δωδεκάεδρο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dodecaedro" title="Dodecaedro – Spanish" lang="es" hreflang="es" data-title="Dodecaedro" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Dekduedro" title="Dekduedro – Esperanto" lang="eo" hreflang="eo" data-title="Dekduedro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Dodekaedro" title="Dodekaedro – Basque" lang="eu" hreflang="eu" data-title="Dodekaedro" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%88%D8%A7%D8%B2%D8%AF%D9%87%E2%80%8C%D9%88%D8%AC%D9%87%DB%8C" title="دوازده‌وجهی – Persian" lang="fa" hreflang="fa" data-title="دوازده‌وجهی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Dod%C3%A9ca%C3%A8dre" title="Dodécaèdre – French" lang="fr" hreflang="fr" data-title="Dodécaèdre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Dodecaedro" title="Dodecaedro – Galician" lang="gl" hreflang="gl" data-title="Dodecaedro" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%AD%EC%9D%B4%EB%A9%B4%EC%B2%B4" title="십이면체 – Korean" lang="ko" hreflang="ko" data-title="십이면체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A1%D5%BD%D5%B6%D5%A5%D6%80%D5%AF%D5%B8%D6%82%D5%A1%D5%B6%D5%AB%D5%BD%D5%BF" title="Տասներկուանիստ – Armenian" lang="hy" hreflang="hy" data-title="Տասներկուանիստ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Dodekaedar" title="Dodekaedar – Croatian" lang="hr" hreflang="hr" data-title="Dodekaedar" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Dodekaedro" title="Dodekaedro – Ido" lang="io" hreflang="io" data-title="Dodekaedro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Dodekahedron" title="Dodekahedron – Indonesian" lang="id" hreflang="id" data-title="Dodekahedron" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dodecaedro" title="Dodecaedro – Italian" lang="it" hreflang="it" data-title="Dodecaedro" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A8%D7%99%D7%A1%D7%A8%D7%95%D7%9F" title="תריסרון – Hebrew" lang="he" hreflang="he" data-title="תריסרון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Dodekay%C3%A8d" title="Dodekayèd – Haitian Creole" lang="ht" hreflang="ht" data-title="Dodekayèd" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Dodecahedron" title="Dodecahedron – Latin" lang="la" hreflang="la" data-title="Dodecahedron" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Dodekaedrs" title="Dodekaedrs – Latvian" lang="lv" hreflang="lv" data-title="Dodekaedrs" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Dodekaedras" title="Dodekaedras – Lithuanian" lang="lt" hreflang="lt" data-title="Dodekaedras" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dodeka%C3%A9der" title="Dodekaéder – Hungarian" lang="hu" hreflang="hu" data-title="Dodekaéder" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B4%D0%B5%D0%BA%D0%B0%D0%B5%D0%B4%D0%B0%D1%80" title="Додекаедар – Macedonian" lang="mk" hreflang="mk" data-title="Додекаедар" 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_twaalfvlak" title="Regelmatig twaalfvlak – Dutch" lang="nl" hreflang="nl" data-title="Regelmatig twaalfvlak" 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/%E5%8D%81%E4%BA%8C%E9%9D%A2%E4%BD%93" title="十二面体 – Japanese" lang="ja" hreflang="ja" data-title="十二面体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Dodekaeder" title="Dodekaeder – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Dodekaeder" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Dodekaeder" title="Dodekaeder – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Dodekaeder" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dodecaedro" title="Dodecaedro – Portuguese" lang="pt" hreflang="pt" data-title="Dodecaedro" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Dodecaedru" title="Dodecaedru – Romanian" lang="ro" hreflang="ro" data-title="Dodecaedru" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Chunka_iskayniyuq_uya" title="Chunka iskayniyuq uya – Quechua" lang="qu" hreflang="qu" data-title="Chunka iskayniyuq uya" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B2%D0%B5%D0%BD%D0%B0%D0%B4%D1%86%D0%B0%D1%82%D0%B8%D0%B3%D1%80%D0%B0%D0%BD%D0%BD%D0%B8%D0%BA%D0%B8" title="Двенадцатигранники – Russian" lang="ru" hreflang="ru" data-title="Двенадцатигранники" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Dodecaedru" title="Dodecaedru – Sicilian" lang="scn" hreflang="scn" data-title="Dodecaedru" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dodecahedron" title="Dodecahedron – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dodecahedron" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Dodekaeder" title="Dodekaeder – Slovenian" lang="sl" hreflang="sl" data-title="Dodekaeder" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%D9%88%D8%A7%D8%B2%D8%AF%DB%95%DA%95%D9%88%D9%88" title="دوازدەڕوو – Central Kurdish" lang="ckb" hreflang="ckb" data-title="دوازدەڕوو" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%BE%D0%B4%D0%B5%D0%BA%D0%B0%D0%B5%D0%B4%D0%B0%D1%80" title="Додекаедар – Serbian" lang="sr" hreflang="sr" data-title="Додекаедар" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dodekaedar" title="Dodekaedar – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Dodekaedar" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Dodekaedri" title="Dodekaedri – Finnish" lang="fi" hreflang="fi" data-title="Dodekaedri" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Dodekaeder" title="Dodekaeder – Swedish" lang="sv" hreflang="sv" data-title="Dodekaeder" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%A9%E0%AF%8D%E0%AE%A9%E0%AE%BF%E0%AE%B0%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AF%81%E0%AE%AE%E0%AF%81%E0%AE%95_%E0%AE%90%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%95%E0%AE%AE%E0%AF%8D" title="பன்னிரண்டுமுக ஐங்கோணகம் – Tamil" lang="ta" hreflang="ta" data-title="பன்னிரண்டுமுக ஐங்கோணகம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%94%D0%BE%D0%B4%D0%B5%D0%BA%D0%B0%D1%8D%D0%B4%D1%80" title="Додекаэдр – Tatar" lang="tt" hreflang="tt" data-title="Додекаэдр" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A3%E0%B8%87%E0%B8%AA%E0%B8%B4%E0%B8%9A%E0%B8%AA%E0%B8%AD%E0%B8%87%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2" title="ทรงสิบสองหน้า – Thai" lang="th" hreflang="th" data-title="ทรงสิบสองหน้า" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/On_iki_y%C3%BCzl%C3%BC" title="On iki yüzlü – Turkish" lang="tr" hreflang="tr" data-title="On iki yüzlü" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B2%D0%B0%D0%BD%D0%B0%D0%B4%D1%86%D1%8F%D1%82%D0%B8%D0%B3%D1%80%D0%B0%D0%BD%D0%BD%D0%B8%D0%BA" title="Дванадцятигранник – Ukrainian" lang="uk" hreflang="uk" data-title="Дванадцятигранник" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%81%E4%BA%8C%E9%9D%A2%E9%AB%94" title="十二面體 – Chinese" lang="zh" hreflang="zh" data-title="十二面體" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://rsk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B4%D0%B5%D0%BA%D0%B0%D0%B5%D0%B4%D0%B5%D1%80" title="Додекаедер – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Додекаедер" data-language-autonym="Руски" 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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">Polyhedron with 12 faces</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">This article is about the three-dimensional shape. Not to be confused with <a href="/wiki/Roman_dodecahedron" title="Roman dodecahedron">Roman dodecahedron</a>.</div> <table class="wikitable floatright" width="320"> <caption>Common dodecahedra </caption> <tbody><tr style="text-align:center;"> <th colspan="4">I<sub>h</sub>, order 120 </th></tr> <tr style="text-align:center; vertical-align:bottom;"> <td><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Regular</a> </td> <td><a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">Small stellated</a> </td> <td><a href="/wiki/Great_dodecahedron" title="Great dodecahedron">Great</a> </td> <td><a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">Great stellated</a> </td></tr> <tr style="text-align:center; vertical-align:bottom;"> <td><span typeof="mw:File"><a href="/wiki/File:Dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/80px-Dodecahedron.png" decoding="async" width="80" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/120px-Dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/160px-Dodecahedron.png 2x" data-file-width="984" data-file-height="951" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Small_stellated_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/80px-Small_stellated_dodecahedron.png" decoding="async" width="80" height="84" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/120px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/160px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Great_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/80px-Great_dodecahedron.png" decoding="async" width="80" height="85" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/120px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/160px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Great_stellated_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/80px-Great_stellated_dodecahedron.png" decoding="async" width="80" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/120px-Great_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/160px-Great_stellated_dodecahedron.png 2x" data-file-width="988" data-file-height="950" /></a></span> </td></tr> <tr> <th>T<sub>h</sub>, order 24 </th> <th>T, order 12 </th> <th>O<sub>h</sub>, order 48 </th> <th>Johnson (J<sub>84</sub>) </th></tr> <tr style="text-align:center; vertical-align:bottom;"> <td><a href="/wiki/Pyritohedron" class="mw-redirect" title="Pyritohedron">Pyritohedron</a> </td> <td><a href="/wiki/Tetartoid" class="mw-redirect" title="Tetartoid">Tetartoid</a> </td> <td><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Rhombic</a> </td> <td><a href="/wiki/Triangular_dodecahedron" class="mw-redirect" title="Triangular dodecahedron">Triangular</a> </td></tr> <tr style="text-align:center; vertical-align:bottom;"> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Pyritohedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Pyritohedron.png/86px-Pyritohedron.png" decoding="async" width="86" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Pyritohedron.png/130px-Pyritohedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Pyritohedron.png/173px-Pyritohedron.png 2x" data-file-width="572" data-file-height="530" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Tetartoid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Tetartoid.png/84px-Tetartoid.png" decoding="async" width="84" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Tetartoid.png/126px-Tetartoid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Tetartoid.png/168px-Tetartoid.png 2x" data-file-width="626" data-file-height="598" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Rhombicdodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/90px-Rhombicdodecahedron.jpg" decoding="async" width="90" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/135px-Rhombicdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/180px-Rhombicdodecahedron.jpg 2x" data-file-width="849" data-file-height="754" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Snub_disphenoid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Snub_disphenoid.png/62px-Snub_disphenoid.png" decoding="async" width="62" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Snub_disphenoid.png/93px-Snub_disphenoid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Snub_disphenoid.png/125px-Snub_disphenoid.png 2x" data-file-width="344" data-file-height="441" /></a></span> </td></tr> <tr align="center"> <th colspan="2">D<sub>4h</sub>, order 16 </th> <th colspan="2">D<sub>3h</sub>, order 12 </th></tr> <tr align="center"> <td><a href="/wiki/Rhombo-hexagonal_dodecahedron" class="mw-redirect" title="Rhombo-hexagonal dodecahedron">Rhombo-hexagonal</a> </td> <td><a href="/wiki/Rhombic_dodecahedron#Parallelohedron" title="Rhombic dodecahedron">Rhombo-square</a> </td> <td><a href="/wiki/Trapezo-rhombic_dodecahedron" title="Trapezo-rhombic dodecahedron">Trapezo-rhombic</a> </td> <td><a href="/wiki/Triangular-rhombic_dodecahedron" class="mw-redirect" title="Triangular-rhombic dodecahedron">Rhombo-triangular</a> </td></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Rhombo-hexagonal_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Rhombo-hexagonal_dodecahedron.png/80px-Rhombo-hexagonal_dodecahedron.png" decoding="async" width="80" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Rhombo-hexagonal_dodecahedron.png/120px-Rhombo-hexagonal_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Rhombo-hexagonal_dodecahedron.png/160px-Rhombo-hexagonal_dodecahedron.png 2x" data-file-width="673" data-file-height="814" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Squared_rhombic_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Squared_rhombic_dodecahedron.png/80px-Squared_rhombic_dodecahedron.png" decoding="async" width="80" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Squared_rhombic_dodecahedron.png/120px-Squared_rhombic_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Squared_rhombic_dodecahedron.png/160px-Squared_rhombic_dodecahedron.png 2x" data-file-width="1606" data-file-height="1991" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Trapezo-rhombic_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Trapezo-rhombic_dodecahedron.png/80px-Trapezo-rhombic_dodecahedron.png" decoding="async" width="80" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Trapezo-rhombic_dodecahedron.png/120px-Trapezo-rhombic_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Trapezo-rhombic_dodecahedron.png/160px-Trapezo-rhombic_dodecahedron.png 2x" data-file-width="787" data-file-height="806" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Triangular_square_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Triangular_square_dodecahedron.png/80px-Triangular_square_dodecahedron.png" decoding="async" width="80" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Triangular_square_dodecahedron.png/120px-Triangular_square_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Triangular_square_dodecahedron.png/160px-Triangular_square_dodecahedron.png 2x" data-file-width="816" data-file-height="933" /></a></span> </td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>dodecahedron</b> (from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a> <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%CE%B4%CF%89%CE%B4%CE%B5%CE%BA%CE%AC%CE%B5%CE%B4%CF%81%CE%BF%CE%BD#Ancient_Greek" class="extiw" title="wikt:δωδεκάεδρον">δωδεκάεδρον</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>dōdekáedron</i></span>)</i>; from <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%CE%B4%CF%8E%CE%B4%CE%B5%CE%BA%CE%B1#Ancient_Greek" class="extiw" title="wikt:δώδεκα">δώδεκα</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>dṓdeka</i></span>)</i> 'twelve' and <i> </i><span lang="grc"><a href="https://en.wiktionary.org/wiki/%E1%BC%95%CE%B4%CF%81%CE%B1#Ancient_Greek" class="extiw" title="wikt:ἕδρα">ἕδρα</a></span><i> (<span title="Ancient Greek transliteration" lang="grc-Latn"><i>hédra</i></span>)</i> 'base, seat, face') or <b>duodecahedron</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is any <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> with twelve flat faces. The most familiar dodecahedron is the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> with regular pentagons as faces, which is a <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>. There are also three <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedron" title="Kepler–Poinsot polyhedron">regular star dodecahedra</a>, which are constructed as <a href="/wiki/Stellation" title="Stellation">stellations</a> of the convex form. All of these have <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a>, order 120. </p><p>Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The <a href="#Pyritohedron">pyritohedron</a>, a common crystal form in <a href="/wiki/Pyrite" title="Pyrite">pyrite</a>, has <a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">pyritohedral symmetry</a>, while the <a href="#Tetartoid">tetartoid</a> has <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">tetrahedral symmetry</a>. </p><p>The <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> can be seen as a limiting case of the pyritohedron, and it has <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a>. The <a href="/wiki/Elongated_dodecahedron" title="Elongated dodecahedron">elongated dodecahedron</a> and <a href="/wiki/Trapezo-rhombic_dodecahedron" title="Trapezo-rhombic dodecahedron">trapezo-rhombic dodecahedron</a> variations, along with the rhombic dodecahedra, are <a href="/wiki/Space-filling_polyhedra" class="mw-redirect" title="Space-filling polyhedra">space-filling</a>. There are numerous <a href="#Other_dodecahedra">other dodecahedra</a>. </p><p>While the regular dodecahedron shares many features with other Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.<sup id="cite_ref-cross_2-0" class="reference"><a href="#cite_note-cross-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Regular_dodecahedron">Regular dodecahedron</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=1" title="Edit section: Regular dodecahedron"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Regular dodecahedron</a></div> <p>The convex regular dodecahedron is one of the five regular <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> and can be represented by its <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {5, 3}. </p><p>The <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> is the regular <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> {3, 5}, having five equilateral triangles around each vertex. </p> <table class="wikitable" align="center"> <caption>Four kinds of regular dodecahedra </caption> <tbody><tr align="center"> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/155px-Dodecahedron.png" decoding="async" width="155" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/233px-Dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/310px-Dodecahedron.png 2x" data-file-width="984" data-file-height="951" /></a></span><br />Convex <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Small_stellated_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/142px-Small_stellated_dodecahedron.png" decoding="async" width="142" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/213px-Small_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Small_stellated_dodecahedron.png/284px-Small_stellated_dodecahedron.png 2x" data-file-width="903" data-file-height="953" /></a></span><br /><a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">Small stellated dodecahedron</a> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Great_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/142px-Great_dodecahedron.png" decoding="async" width="142" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/212px-Great_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Great_dodecahedron.png/283px-Great_dodecahedron.png 2x" data-file-width="904" data-file-height="958" /></a></span><br /><a href="/wiki/Great_dodecahedron" title="Great dodecahedron">Great dodecahedron</a> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Great_stellated_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/156px-Great_stellated_dodecahedron.png" decoding="async" width="156" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/234px-Great_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/312px-Great_stellated_dodecahedron.png 2x" data-file-width="988" data-file-height="950" /></a></span><br /><a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">Great stellated dodecahedron</a> </td></tr></tbody></table> <p>The convex regular dodecahedron also has three <a href="/wiki/Stellation" title="Stellation">stellations</a>, all of which are regular star dodecahedra. They form three of the four <a href="/wiki/Kepler%E2%80%93Poinsot_polyhedra" class="mw-redirect" title="Kepler–Poinsot polyhedra">Kepler–Poinsot polyhedra</a>. They are the <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a> {5/2, 5}, the <a href="/wiki/Great_dodecahedron" title="Great dodecahedron">great dodecahedron</a> {5, 5/2}, and the <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a> {5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a> {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or <a href="/wiki/Pentagram" title="Pentagram">pentagrammic</a> faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same <a href="/wiki/Abstract_polytope" title="Abstract polytope">abstract regular polyhedron</a>; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron. </p> <div class="mw-heading mw-heading2"><h2 id="Other_pentagonal_dodecahedra">Other pentagonal dodecahedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=2" title="Edit section: Other pentagonal dodecahedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>, two important dodecahedra can occur as crystal forms in some <a href="/wiki/Crystallographic_point_groups" class="mw-redirect" title="Crystallographic point groups">symmetry classes</a> of the <a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic crystal system</a> that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with <a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">pyritohedral symmetry</a>, and the <a href="/wiki/Tetartoid" class="mw-redirect" title="Tetartoid">tetartoid</a> with <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">tetrahedral symmetry</a>: </p> <div class="mw-heading mw-heading3"><h3 id="Pyritohedron">Pyritohedron</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=3" title="Edit section: Pyritohedron"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable floatright" style="width:260px;"> <tbody><tr> <th style="background:#e7dcc3;" colspan="2">Pyritohedron </th></tr> <tr> <td style="text-align:center;" colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_pyritohedron_transparent_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Polyhedron_pyritohedron_transparent_max.png/250px-Polyhedron_pyritohedron_transparent_max.png" decoding="async" width="250" height="276" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Polyhedron_pyritohedron_transparent_max.png/375px-Polyhedron_pyritohedron_transparent_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Polyhedron_pyritohedron_transparent_max.png/500px-Polyhedron_pyritohedron_transparent_max.png 2x" data-file-width="3260" data-file-height="3600" /></a></span><br /><small>(See <a href="https://commons.wikimedia.org/wiki/File:Polyhedron_pyritohedron_transparent_max.gif" class="extiw" title="c:File:Polyhedron pyritohedron transparent max.gif">here</a> for a rotating model.)</small> </td></tr> <tr> <td style="background:#e7dcc3;">Face polygon</td> <td><a href="/wiki/Pentagon" title="Pentagon">isosceles pentagon</a> </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagrams</a></td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span><br /><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Face_(geometry)" title="Face (geometry)">Faces</a></td> <td>12 </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></td> <td>30 (6 + 24) </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></td> <td>20 (8 + 12) </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/List_of_spherical_symmetry_groups#Polyhedral_sym" title="List of spherical symmetry groups">Symmetry group</a></td> <td><a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">T<sub>h</sub></a>, [4,3<sup>+</sup>], (3*2), order 24 </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Point_groups_in_three_dimensions#Rotation_groups" title="Point groups in three dimensions">Rotation group</a></td> <td><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">T</a>, [3,3]<sup>+</sup>, (332), order 12 </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></td> <td><a href="/wiki/Pseudoicosahedron" class="mw-redirect" title="Pseudoicosahedron">Pseudoicosahedron</a> </td></tr> <tr> <td style="background:#e7dcc3;">Properties</td> <td><a href="/wiki/Face_transitive" class="mw-redirect" title="Face transitive">face transitive</a> </td></tr> <tr align="center"> <td colspan="2"><a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">Net</a><br /><span typeof="mw:File"><a href="/wiki/File:Pyritohedron_flat.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Pyritohedron_flat.png/150px-Pyritohedron_flat.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Pyritohedron_flat.png/225px-Pyritohedron_flat.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Pyritohedron_flat.png/300px-Pyritohedron_flat.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td></tr></tbody></table> <p>A <b>pyritohedron</b> is a dodecahedron with <a href="/wiki/Pyritohedral_symmetry" class="mw-redirect" title="Pyritohedral symmetry">pyritohedral</a> (T<sub>h</sub>) symmetry. Like the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a>, it has twelve identical <a href="/wiki/Pentagon" title="Pentagon">pentagonal</a> faces, with three meeting in each of the 20 vertices (see figure).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> are three mutually perpendicular twofold axes and four threefold axes. </p><p>Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral <a href="/wiki/Pyrite" title="Pyrite">pyrite</a>, and it may be an inspiration for the discovery of the regular <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a> form. The true regular dodecahedron can occur as a shape for <a href="/wiki/Quasicrystal" title="Quasicrystal">quasicrystals</a> (such as <a href="/wiki/Holmium%E2%80%93magnesium%E2%80%93zinc_quasicrystal" title="Holmium–magnesium–zinc quasicrystal">holmium–magnesium–zinc quasicrystal</a>) with <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a>, which includes true fivefold rotation axes. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Modell_eines_Kristalls_des_Minerals_Pyrit_(Eisernes_Kreuz)_-Krantz_375-_(2),_crop.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Modell_eines_Kristalls_des_Minerals_Pyrit_%28Eisernes_Kreuz%29_-Krantz_375-_%282%29%2C_crop.jpg/190px-Modell_eines_Kristalls_des_Minerals_Pyrit_%28Eisernes_Kreuz%29_-Krantz_375-_%282%29%2C_crop.jpg" decoding="async" width="190" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Modell_eines_Kristalls_des_Minerals_Pyrit_%28Eisernes_Kreuz%29_-Krantz_375-_%282%29%2C_crop.jpg/285px-Modell_eines_Kristalls_des_Minerals_Pyrit_%28Eisernes_Kreuz%29_-Krantz_375-_%282%29%2C_crop.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Modell_eines_Kristalls_des_Minerals_Pyrit_%28Eisernes_Kreuz%29_-Krantz_375-_%282%29%2C_crop.jpg/380px-Modell_eines_Kristalls_des_Minerals_Pyrit_%28Eisernes_Kreuz%29_-Krantz_375-_%282%29%2C_crop.jpg 2x" data-file-width="2762" data-file-height="1920" /></a><figcaption>Dual positions in pyrite <a href="/wiki/Crystal_model" title="Crystal model">crystal models</a></figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Crystal_pyrite">Crystal pyrite</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=4" title="Edit section: Crystal pyrite"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The name <i>crystal pyrite</i> comes from one of the two common <a href="/wiki/Crystal_habit" title="Crystal habit">crystal habits</a> shown by <a href="/wiki/Pyrite" title="Pyrite">pyrite</a> (the other one being the <a href="/wiki/Cube" title="Cube">cube</a>). In pyritohedral pyrite, the faces have a <a href="/wiki/Miller_index" title="Miller index">Miller index</a> of (210), which means that the <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature). </p><p><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 {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Height</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Long side</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f41c51231bebe0a9703575d38b6f73be4b8f32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.033ex; height:5.843ex;" alt="{\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}}"></span> </p><p><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 {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Width</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Long side</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc3e37d2a85c5f2647ff1aaa874ab30303ff1a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.84ex; height:5.176ex;" alt="{\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}}"></span> </p><p><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 {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Short sides</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>7</mn> <mn>12</mn> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Long side</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ee02eb8cc1920926996dcd29653c2976d663d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.75ex; height:6.176ex;" alt="{\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}}"></span> </p> <table> <tbody><tr> <td> <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 tleft"><div class="thumbinner multiimageinner" style="width:312px;max-width:312px"><div class="trow"><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:142px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Pyrite-184681.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Pyrite-184681.jpg/152px-Pyrite-184681.jpg" decoding="async" width="152" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Pyrite-184681.jpg/228px-Pyrite-184681.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Pyrite-184681.jpg/304px-Pyrite-184681.jpg 2x" data-file-width="400" data-file-height="375" /></a></span></div></div><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:142px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Pyrite-193871_angles.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Pyrite-193871_angles.jpg/152px-Pyrite-193871_angles.jpg" decoding="async" width="152" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Pyrite-193871_angles.jpg/228px-Pyrite-193871_angles.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Pyrite-193871_angles.jpg/304px-Pyrite-193871_angles.jpg 2x" data-file-width="750" data-file-height="700" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Natural pyrite (with face angles on the right)</div></div></div></div> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Cartesian_coordinates">Cartesian coordinates</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=5" title="Edit section: Cartesian coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The eight vertices of a cube have the coordinates (±1, ±1, ±1). </p><p>The coordinates of the 12 additional vertices are <big>(</big>0, ±(1 + <i>h</i>), ±(1 − <i>h</i><sup>2</sup>)<big>)</big>, <big>(</big>±(1 + <i>h</i>), ±(1 − <i>h</i><sup>2</sup>), 0<big>)</big> and <big>(</big>±(1 − <i>h</i><sup>2</sup>), 0, ±(1 + <i>h</i>)<big>)</big>. </p><p><i>h</i> is the height of the <a href="/wiki/Wedge_(geometry)" title="Wedge (geometry)">wedge</a>-shaped "roof" above the faces of that cube with edge length 2. </p><p>An important case is <i>h</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the <a href="/wiki/Weaire%E2%80%93Phelan_structure" title="Weaire–Phelan structure">Weaire–Phelan structure</a>). </p><p>Another one is <i>h</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><a href="/wiki/Golden_ratio" title="Golden ratio">φ</a></span></span>⁠</span> = 0.618... for the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a>. See section <i><a href="#Geometric_freedom">Geometric freedom</a></i> for other cases. </p><p>Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the <a href="/wiki/Compound_of_two_dodecahedra" class="mw-redirect" title="Compound of two dodecahedra">compound of two dodecahedra</a>. </p> <table> <tbody><tr> <td> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:432px;max-width:432px"><div class="trow"><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_pyritohedron_from_yellow_max.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Polyhedron_pyritohedron_from_yellow_max.png/140px-Polyhedron_pyritohedron_from_yellow_max.png" decoding="async" width="140" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Polyhedron_pyritohedron_from_yellow_max.png/210px-Polyhedron_pyritohedron_from_yellow_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Polyhedron_pyritohedron_from_yellow_max.png/280px-Polyhedron_pyritohedron_from_yellow_max.png 2x" data-file-width="1752" data-file-height="1798" /></a></span></div></div><div class="tsingle" style="width:146px;max-width:146px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_pyritohedron_from_red_max.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Polyhedron_pyritohedron_from_red_max.png/144px-Polyhedron_pyritohedron_from_red_max.png" decoding="async" width="144" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Polyhedron_pyritohedron_from_red_max.png/216px-Polyhedron_pyritohedron_from_red_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Polyhedron_pyritohedron_from_red_max.png/288px-Polyhedron_pyritohedron_from_red_max.png 2x" data-file-width="1664" data-file-height="1662" /></a></span></div></div><div class="tsingle" style="width:138px;max-width:138px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_pyritohedron_from_blue_max.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Polyhedron_pyritohedron_from_blue_max.png/136px-Polyhedron_pyritohedron_from_blue_max.png" decoding="async" width="136" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Polyhedron_pyritohedron_from_blue_max.png/204px-Polyhedron_pyritohedron_from_blue_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Polyhedron_pyritohedron_from_blue_max.png/272px-Polyhedron_pyritohedron_from_blue_max.png 2x" data-file-width="1662" data-file-height="1760" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Orthographic projections of the pyritohedron with <i>h</i> = 1/2</div></div></div></div> </td> <td> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:270px;max-width:270px"><div class="trow"><div class="tsingle" style="width:132px;max-width:132px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_pyritohedron_max.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Polyhedron_pyritohedron_max.png/130px-Polyhedron_pyritohedron_max.png" decoding="async" width="130" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Polyhedron_pyritohedron_max.png/195px-Polyhedron_pyritohedron_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Polyhedron_pyritohedron_max.png/260px-Polyhedron_pyritohedron_max.png 2x" data-file-width="3252" data-file-height="3584" /></a></span></div></div><div class="tsingle" style="width:134px;max-width:134px"><div class="thumbimage" style="height:143px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_12_pyritohedral_max.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Polyhedron_12_pyritohedral_max.png/132px-Polyhedron_12_pyritohedral_max.png" decoding="async" width="132" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Polyhedron_12_pyritohedral_max.png/198px-Polyhedron_12_pyritohedral_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Polyhedron_12_pyritohedral_max.png/264px-Polyhedron_12_pyritohedral_max.png 2x" data-file-width="3652" data-file-height="3960" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Heights 1/2 and 1/<a href="/wiki/Golden_Ratio" class="mw-redirect" title="Golden Ratio"><i>φ</i></a></div></div></div></div> </td> <td> </td></tr></tbody></table> <table class="wikitable collapsible collapsed" style="text-align: center;"> <tbody><tr> <th colspan="2">Animations </th></tr> <tr style="background-color: white;"> <td style="width: 350px;"><span typeof="mw:File"><a href="/wiki/File:Endo-dodecahedron_honeycomb.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Endo-dodecahedron_honeycomb.gif/200px-Endo-dodecahedron_honeycomb.gif" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Endo-dodecahedron_honeycomb.gif/300px-Endo-dodecahedron_honeycomb.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Endo-dodecahedron_honeycomb.gif/400px-Endo-dodecahedron_honeycomb.gif 2x" data-file-width="800" data-file-height="799" /></a></span> </td> <td style="width: 350px;"><span typeof="mw:File"><a href="/wiki/File:Pyritohedron_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Pyritohedron_animation.gif/200px-Pyritohedron_animation.gif" decoding="async" width="200" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Pyritohedron_animation.gif/300px-Pyritohedron_animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/1/19/Pyritohedron_animation.gif 2x" data-file-width="400" data-file-height="437" /></a></span> </td></tr> <tr> <td><a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">Honeycomb</a> of alternating convex and concave pyritohedra with heights between ±<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><a href="/wiki/Golden_ratio" title="Golden ratio">φ</a></span></span>⁠</span> </td> <td>Heights between 0 (cube)<br />and 1 (rhombic dodecahedron) </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Geometric_freedom">Geometric freedom</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=6" title="Edit section: Geometric freedom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The pyritohedron has a geometric degree of freedom with <a href="/wiki/Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting cases</a> of a cubic <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> at one limit of collinear edges, and a <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal. </p><p>It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The <i>endo-dodecahedron</i> is concave and equilateral; it can tessellate space with the convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a> where all edges and angles are equal again, and the faces have been distorted into regular <a href="/wiki/Pentagram_(geometry)" class="mw-redirect" title="Pentagram (geometry)">pentagrams</a>. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces. </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th colspan="8">Special cases of the pyritohedron </th></tr> <tr> <td colspan="8">Versions with equal absolute values and opposing signs form a honeycomb together. (Compare <a href="/wiki/File:Endo-dodecahedron_honeycomb.gif" title="File:Endo-dodecahedron honeycomb.gif">this animation</a>.)<br />The ratio shown is that of edge lengths, namely those in a set of 24 (touching cube vertices) to those in a set of 6 (corresponding to cube faces). </td></tr> <tr> <th>Ratio </th> <th>1 : 1 </th> <th>0 : 1 </th> <th>1 : 1 </th> <th>2 : 1 </th> <th>1 : 1 </th> <th>0 : 1 </th> <th>1 : 1 </th></tr> <tr> <th rowspan="2"><i>h</i> </th> <th>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span> + 1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </th> <th rowspan="2">−1 </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">−<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span> + 1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </th> <th rowspan="2">0 </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span> − 1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </th> <th rowspan="2">1 </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span> + 1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </th></tr> <tr> <th>−1.618... </th> <th>−0.618... </th> <th>0.618... </th> <th>1.618... </th></tr> <tr style="text-align: center; vertical-align: top;"> <th style="vertical-align: middle;">Image </th> <td><span typeof="mw:File"><a href="/wiki/File:Great_stellated_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/120px-Great_stellated_dodecahedron.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/180px-Great_stellated_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Great_stellated_dodecahedron.png/240px-Great_stellated_dodecahedron.png 2x" data-file-width="988" data-file-height="950" /></a></span><br />Regular star, <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a>, with regular <a href="/wiki/Pentagram" title="Pentagram">pentagram</a> faces </td> <td><span typeof="mw:File"><a href="/wiki/File:Degenerate-pyritohedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Degenerate-pyritohedron.png/120px-Degenerate-pyritohedron.png" decoding="async" width="120" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Degenerate-pyritohedron.png/180px-Degenerate-pyritohedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Degenerate-pyritohedron.png/240px-Degenerate-pyritohedron.png 2x" data-file-width="729" data-file-height="797" /></a></span><br />Degenerate, 12 vertices in the center </td> <td><span typeof="mw:File"><a href="/wiki/File:Concave_pyritohedral_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Concave_pyritohedral_dodecahedron.png/120px-Concave_pyritohedral_dodecahedron.png" decoding="async" width="120" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Concave_pyritohedral_dodecahedron.png/180px-Concave_pyritohedral_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Concave_pyritohedral_dodecahedron.png/240px-Concave_pyritohedral_dodecahedron.png 2x" data-file-width="454" data-file-height="476" /></a></span><br />The concave equilateral dodecahedron, called an <i>endo-dodecahedron</i>. <sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Image should be replaced by one with the specified height. (October 2020)">clarification needed</span></a></i>]</sup> </td> <td><span typeof="mw:File"><a href="/wiki/File:Pyritohedron_cube.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Pyritohedron_cube.png/120px-Pyritohedron_cube.png" decoding="async" width="120" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Pyritohedron_cube.png/180px-Pyritohedron_cube.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Pyritohedron_cube.png/240px-Pyritohedron_cube.png 2x" data-file-width="719" data-file-height="771" /></a></span><br />A <a href="/wiki/Cube" title="Cube">cube</a> can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. </td> <td><span typeof="mw:File"><a href="/wiki/File:Dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/120px-Dodecahedron.png" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/180px-Dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/240px-Dodecahedron.png 2x" data-file-width="984" data-file-height="951" /></a></span><br />A regular dodecahedron is an intermediate case with equal edge lengths. </td> <td><span typeof="mw:File"><a href="/wiki/File:Rhombicdodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/120px-Rhombicdodecahedron.jpg" decoding="async" width="120" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/180px-Rhombicdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/240px-Rhombicdodecahedron.jpg 2x" data-file-width="849" data-file-height="754" /></a></span><br />A <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> is a degenerate case with the 6 crossedges reduced to length zero. </td> <td><span typeof="mw:File"><a href="/wiki/File:Exo-dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Exo-dodecahedron.png/120px-Exo-dodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Exo-dodecahedron.png/180px-Exo-dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Exo-dodecahedron.png/240px-Exo-dodecahedron.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br />Self-intersecting equilateral dodecahedron </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Tetartoid">Tetartoid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=7" title="Edit section: Tetartoid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable floatright" style="width:260px;"> <tbody><tr> <th style="background:#e7dcc3;" colspan="2">Tetartoid<br />Tetragonal pentagonal dodecahedron </th></tr> <tr> <td style="text-align:center;" colspan="2"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_perspective.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Tetartoid_perspective.png/250px-Tetartoid_perspective.png" decoding="async" width="250" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Tetartoid_perspective.png/375px-Tetartoid_perspective.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Tetartoid_perspective.png/500px-Tetartoid_perspective.png 2x" data-file-width="3471" data-file-height="3630" /></a></span><br /><small>(See <a href="https://commons.wikimedia.org/wiki/File:Tetartoid_perspective.gif" class="extiw" title="c:File:Tetartoid perspective.gif">here</a> for a rotating model.)</small> </td></tr> <tr> <td style="background:#e7dcc3;">Face polygon</td> <td><a href="/wiki/Pentagon" title="Pentagon">irregular pentagon</a> </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">Conway notation</a></td> <td>gT </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Face_(geometry)" title="Face (geometry)">Faces</a></td> <td>12 </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></td> <td>30 (6+12+12) </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></td> <td>20 (4+4+12) </td></tr> <tr> <td style="background:#e7dcc3;"><a href="/wiki/List_of_spherical_symmetry_groups#Polyhedral_sym" title="List of spherical symmetry groups">Symmetry group</a></td> <td><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">T</a>, [3,3]<sup>+</sup>, (332), order 12 </td></tr> <tr> <td style="background:#e7dcc3;">Properties</td> <td><a href="/wiki/Convex_set" title="Convex set">convex</a>, <a href="/wiki/Face_transitive" class="mw-redirect" title="Face transitive">face transitive</a> </td></tr></tbody></table> <p>A <b>tetartoid</b> (also <b>tetragonal pentagonal dodecahedron</b>, <b>pentagon-tritetrahedron</b>, and <b>tetrahedric pentagon dodecahedron</b>) is a dodecahedron with chiral <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">tetrahedral symmetry</a> (T). Like the <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a>, it has twelve identical <a href="/wiki/Pentagon" title="Pentagon">pentagonal</a> faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. </p><p>Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The mineral <a href="/wiki/Cobaltite" title="Cobaltite">cobaltite</a> can have this symmetry form.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Abstractions sharing the solid's <a href="/wiki/Topology" title="Topology">topology</a> and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In <a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">Conway polyhedron notation</a> this is a gyro tetrahedron.) </p> <table> <tbody><tr> <td> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:433px;max-width:433px"><div class="trow"><div class="tsingle" style="width:141px;max-width:141px"><div class="thumbimage" style="height:138px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_from_red.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Tetartoid_from_red.png/139px-Tetartoid_from_red.png" decoding="async" width="139" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Tetartoid_from_red.png/209px-Tetartoid_from_red.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Tetartoid_from_red.png/278px-Tetartoid_from_red.png 2x" data-file-width="1924" data-file-height="1924" /></a></span></div></div><div class="tsingle" style="width:143px;max-width:143px"><div class="thumbimage" style="height:138px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_from_green.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Tetartoid_from_green.png/141px-Tetartoid_from_green.png" decoding="async" width="141" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Tetartoid_from_green.png/212px-Tetartoid_from_green.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Tetartoid_from_green.png/282px-Tetartoid_from_green.png 2x" data-file-width="2010" data-file-height="1984" /></a></span></div></div><div class="tsingle" style="width:143px;max-width:143px"><div class="thumbimage" style="height:138px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_from_yellow.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Tetartoid_from_yellow.png/141px-Tetartoid_from_yellow.png" decoding="async" width="141" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Tetartoid_from_yellow.png/212px-Tetartoid_from_yellow.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Tetartoid_from_yellow.png/282px-Tetartoid_from_yellow.png 2x" data-file-width="2010" data-file-height="1984" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Orthographic projections from 2- and 3-fold axes</div></div></div></div> </td> <td> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage" style="height:138px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_cube.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Tetartoid_cube.png/140px-Tetartoid_cube.png" decoding="async" width="140" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Tetartoid_cube.png/210px-Tetartoid_cube.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Tetartoid_cube.png/280px-Tetartoid_cube.png 2x" data-file-width="3657" data-file-height="3619" /></a></span></div></div><div class="tsingle" style="width:146px;max-width:146px"><div class="thumbimage" style="height:138px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_tetrahedron.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Tetartoid_tetrahedron.png/144px-Tetartoid_tetrahedron.png" decoding="async" width="144" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Tetartoid_tetrahedron.png/216px-Tetartoid_tetrahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Tetartoid_tetrahedron.png/288px-Tetartoid_tetrahedron.png 2x" data-file-width="3657" data-file-height="3531" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Cubic and tetrahedral form</div></div></div></div> </td> <td> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cobaltite-d05-67a.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Cobaltite-d05-67a.jpg/143px-Cobaltite-d05-67a.jpg" decoding="async" width="143" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Cobaltite-d05-67a.jpg/215px-Cobaltite-d05-67a.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Cobaltite-d05-67a.jpg/286px-Cobaltite-d05-67a.jpg 2x" data-file-width="402" data-file-height="398" /></a><figcaption><a href="/wiki/Cobaltite" title="Cobaltite">Cobaltite</a></figcaption></figure> </td></tr></tbody></table> <table class="wikitable collapsible collapsed"> <tbody><tr> <th>Relationship to the dyakis dodecahedron </th></tr> <tr> <td style="width: 760px;"> <p>A tetartoid can be created by enlarging 12 of the 24 faces of a <a href="/wiki/Dyakis_dodecahedron" title="Dyakis dodecahedron">dyakis dodecahedron</a>. (The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the <a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a>.) </p> <table> <tbody><tr> <td> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:543px;max-width:543px"><div class="trow"><div class="tsingle" style="width:179px;max-width:179px"><div class="thumbimage" style="height:176px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_dark_vertical_(with_traces_of_dyakis_12).png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Tetartoid_dark_vertical_%28with_traces_of_dyakis_12%29.png/177px-Tetartoid_dark_vertical_%28with_traces_of_dyakis_12%29.png" decoding="async" width="177" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Tetartoid_dark_vertical_%28with_traces_of_dyakis_12%29.png/266px-Tetartoid_dark_vertical_%28with_traces_of_dyakis_12%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Tetartoid_dark_vertical_%28with_traces_of_dyakis_12%29.png/354px-Tetartoid_dark_vertical_%28with_traces_of_dyakis_12%29.png 2x" data-file-width="4000" data-file-height="4000" /></a></span></div></div><div class="tsingle" style="width:179px;max-width:179px"><div class="thumbimage" style="height:176px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Disdyakis_12_untruncated_to_dyakis_12_vertical.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Disdyakis_12_untruncated_to_dyakis_12_vertical.png/177px-Disdyakis_12_untruncated_to_dyakis_12_vertical.png" decoding="async" width="177" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Disdyakis_12_untruncated_to_dyakis_12_vertical.png/266px-Disdyakis_12_untruncated_to_dyakis_12_vertical.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Disdyakis_12_untruncated_to_dyakis_12_vertical.png/354px-Disdyakis_12_untruncated_to_dyakis_12_vertical.png 2x" data-file-width="4000" data-file-height="4000" /></a></span></div></div><div class="tsingle" style="width:179px;max-width:179px"><div class="thumbimage" style="height:176px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetartoid_light_vertical_(with_traces_of_dyakis_12).png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Tetartoid_light_vertical_%28with_traces_of_dyakis_12%29.png/177px-Tetartoid_light_vertical_%28with_traces_of_dyakis_12%29.png" decoding="async" width="177" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Tetartoid_light_vertical_%28with_traces_of_dyakis_12%29.png/266px-Tetartoid_light_vertical_%28with_traces_of_dyakis_12%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Tetartoid_light_vertical_%28with_traces_of_dyakis_12%29.png/354px-Tetartoid_light_vertical_%28with_traces_of_dyakis_12%29.png 2x" data-file-width="4000" data-file-height="4000" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="/wiki/Chirality" title="Chirality">Chiral</a> tetartoids based on the dyakis dodecahedron in the middle</div></div></div></div> </td> <td> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Crystal_model_of_tetartoid_around_dyakis_dodecahedron_(mirrored).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Crystal_model_of_tetartoid_around_dyakis_dodecahedron_%28mirrored%29.jpg/155px-Crystal_model_of_tetartoid_around_dyakis_dodecahedron_%28mirrored%29.jpg" decoding="async" width="155" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Crystal_model_of_tetartoid_around_dyakis_dodecahedron_%28mirrored%29.jpg/233px-Crystal_model_of_tetartoid_around_dyakis_dodecahedron_%28mirrored%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Crystal_model_of_tetartoid_around_dyakis_dodecahedron_%28mirrored%29.jpg/310px-Crystal_model_of_tetartoid_around_dyakis_dodecahedron_%28mirrored%29.jpg 2x" data-file-width="2142" data-file-height="2502" /></a><figcaption>Crystal model</figcaption></figure> </td></tr></tbody></table> <p>The <a href="/wiki/Crystal_model" title="Crystal model">crystal model</a> on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Cartesian_coordinates_2">Cartesian coordinates</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=8" title="Edit section: Cartesian coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following points are vertices of a tetartoid pentagon under <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">tetrahedral symmetry</a>: </p> <dl><dd>(<i>a</i>, <i>b</i>, <i>c</i>); (−<i>a</i>, −<i>b</i>, <i>c</i>); (−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i><sub>1</sub></span></span>⁠</span>, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i><sub>1</sub></span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i><sub>1</sub></span></span>⁠</span>); (−<i>c</i>, −<i>a</i>, <i>b</i>); (−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i><sub>2</sub></span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i><sub>2</sub></span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den"><i>d</i><sub>2</sub></span></span>⁠</span>),</dd></dl> <p>under the following conditions:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="nowrap">0 ≤ <i>a</i> ≤ <i>b</i> ≤ <i>c</i></span>,</dd> <dd><i>n</i> = <i>a</i><sup>2</sup><i>c</i> − <i>bc</i><sup>2</sup>,</dd> <dd><i>d</i><sub>1</sub> = <i>a</i><sup>2</sup> − <i>ab</i> + <i>b</i><sup>2</sup> + <i>ac</i> − 2<i>bc</i>,</dd> <dd><i>d</i><sub>2</sub> = <i>a</i><sup>2</sup> + <i>ab</i> + <i>b</i><sup>2</sup> − <i>ac</i> − 2<i>bc</i>,</dd> <dd><span class="nowrap"><i>nd</i><sub>1</sub><i>d</i><sub>2</sub> ≠ 0</span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Geometric_freedom_2">Geometric freedom</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=9" title="Edit section: Geometric freedom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> is a tetartoid with more than the required symmetry. The <a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a> is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.) </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th colspan="8">Tetartoid variations from <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a> to <a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a> </th></tr> <tr style="background-color: white;"> <td><span typeof="mw:File"><a href="/wiki/File:Dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/140px-Dodecahedron.png" decoding="async" width="140" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/210px-Dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Dodecahedron.png/280px-Dodecahedron.png 2x" data-file-width="984" data-file-height="951" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tetartoid-010.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Tetartoid-010.png/150px-Tetartoid-010.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Tetartoid-010.png/225px-Tetartoid-010.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Tetartoid-010.png/300px-Tetartoid-010.png 2x" data-file-width="2000" data-file-height="2000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tetartoid-020.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Tetartoid-020.png/150px-Tetartoid-020.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Tetartoid-020.png/225px-Tetartoid-020.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Tetartoid-020.png/300px-Tetartoid-020.png 2x" data-file-width="2000" data-file-height="2000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tetartoid-040.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Tetartoid-040.png/150px-Tetartoid-040.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Tetartoid-040.png/225px-Tetartoid-040.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Tetartoid-040.png/300px-Tetartoid-040.png 2x" data-file-width="2000" data-file-height="2000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tetartoid-060.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Tetartoid-060.png/150px-Tetartoid-060.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Tetartoid-060.png/225px-Tetartoid-060.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Tetartoid-060.png/300px-Tetartoid-060.png 2x" data-file-width="2000" data-file-height="2000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tetartoid-080.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Tetartoid-080.png/150px-Tetartoid-080.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Tetartoid-080.png/225px-Tetartoid-080.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Tetartoid-080.png/300px-Tetartoid-080.png 2x" data-file-width="2000" data-file-height="2000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Tetartoid-095.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Tetartoid-095.png/150px-Tetartoid-095.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Tetartoid-095.png/225px-Tetartoid-095.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Tetartoid-095.png/300px-Tetartoid-095.png 2x" data-file-width="2000" data-file-height="2000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Triakistetrahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Triakistetrahedron.jpg/100px-Triakistetrahedron.jpg" decoding="async" width="100" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Triakistetrahedron.jpg/150px-Triakistetrahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Triakistetrahedron.jpg/200px-Triakistetrahedron.jpg 2x" data-file-width="681" data-file-height="766" /></a></span> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Dual_of_triangular_gyrobianticupola">Dual of triangular gyrobianticupola</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=10" title="Edit section: Dual of triangular gyrobianticupola"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular <a href="/wiki/Anticupola" class="mw-redirect" title="Anticupola">anticupola</a> connected base-to-base, called a <i>triangular gyrobianticupola.</i> It has D<sub>3d</sub> symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal cross-section and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not match. </p> <dl><dd><span typeof="mw:File"><a href="/wiki/File:Dual_triangular_gyrobianticupola.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Dual_triangular_gyrobianticupola.png/160px-Dual_triangular_gyrobianticupola.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Dual_triangular_gyrobianticupola.png/240px-Dual_triangular_gyrobianticupola.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Dual_triangular_gyrobianticupola.png/320px-Dual_triangular_gyrobianticupola.png 2x" data-file-width="1200" data-file-height="1200" /></a></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Rhombic_dodecahedron">Rhombic dodecahedron</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=11" title="Edit section: Rhombic dodecahedron"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Rhombicdodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/160px-Rhombicdodecahedron.jpg" decoding="async" width="160" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/240px-Rhombicdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/320px-Rhombicdodecahedron.jpg 2x" data-file-width="849" data-file-height="754" /></a><figcaption>Rhombic dodecahedron</figcaption></figure> <p>The <i><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></i> is a <a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a> with twelve rhombic faces and octahedral symmetry. It is dual to the <a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">quasiregular</a> <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a> (an <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solid</a>) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space. </p><p>The <i>rhombic dodecahedron</i> can be seen as a degenerate <a href="/wiki/Pyritohedron" class="mw-redirect" title="Pyritohedron">pyritohedron</a> where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces. </p><p>The rhombic dodecahedron has several <a href="/wiki/Stellation" title="Stellation">stellations</a>, the <a href="/wiki/First_stellation_of_rhombic_dodecahedron" class="mw-redirect" title="First stellation of rhombic dodecahedron">first of which</a> is also a <a href="/wiki/Rhombic_dodecahedron#Parallelohedron" title="Rhombic dodecahedron">parallelohedral spacefiller</a>. </p><p>Another important rhombic dodecahedron, the <a href="/wiki/Bilinski_dodecahedron" title="Bilinski dodecahedron">Bilinski dodecahedron</a>, has twelve faces congruent to those of the <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a>, i.e. the diagonals are in the ratio of the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>. It is also a <a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a> and was described by <a href="/wiki/Stanko_Bilinski" title="Stanko Bilinski">Bilinski</a> in 1960.<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> This figure is another spacefiller, and can also occur in non-periodic <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">spacefillings</a> along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Other_dodecahedra">Other dodecahedra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=12" title="Edit section: Other dodecahedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 6,384,634 topologically distinct <i>convex</i> dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) </p><p>Topologically distinct dodecahedra (excluding pentagonal and rhombic forms) </p> <ul><li>Uniform polyhedra: <ul><li><a href="/wiki/Decagonal_prism" class="mw-redirect" title="Decagonal prism">Decagonal prism</a> – 10 squares, 2 decagons, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>10h</sub></a> symmetry, order 40.</li> <li><a href="/wiki/Pentagonal_antiprism" title="Pentagonal antiprism">Pentagonal antiprism</a> – 10 equilateral triangles, 2 pentagons, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>5d</sub></a> symmetry, order 20</li></ul></li> <li><a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a> (regular faced): <ul><li><a href="/wiki/Pentagonal_cupola" title="Pentagonal cupola">Pentagonal cupola</a> – 5 triangles, 5 squares, 1 pentagon, 1 decagon, <a href="/wiki/Cyclic_symmetry" class="mw-redirect" title="Cyclic symmetry">C<sub>5v</sub></a> symmetry, order 10</li> <li><a href="/wiki/Snub_disphenoid" title="Snub disphenoid">Snub disphenoid</a> – 12 triangles, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>2d</sub></a>, order 8</li> <li><a href="/wiki/Elongated_square_dipyramid" class="mw-redirect" title="Elongated square dipyramid">Elongated square dipyramid</a> – 8 triangles and 4 squares, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>4h</sub></a> symmetry, order 16</li> <li><a href="/wiki/Metabidiminished_icosahedron" title="Metabidiminished icosahedron">Metabidiminished icosahedron</a> – 10 triangles and 2 pentagons, <a href="/wiki/Cyclic_symmetry" class="mw-redirect" title="Cyclic symmetry">C<sub>2v</sub></a> symmetry, order 4</li></ul></li> <li>Congruent irregular faced: (<a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>) <ul><li><a href="/wiki/Hexagonal_bipyramid" title="Hexagonal bipyramid">Hexagonal bipyramid</a> – 12 isosceles <a href="/wiki/Triangle" title="Triangle">triangles</a>, dual of <a href="/wiki/Hexagonal_prism" title="Hexagonal prism">hexagonal prism</a>, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>6h</sub></a> symmetry, order 24</li> <li><a href="/wiki/Hexagonal_trapezohedron" title="Hexagonal trapezohedron">Hexagonal trapezohedron</a> – 12 <a href="/wiki/Kite_(geometry)" title="Kite (geometry)">kites</a>, dual of <a href="/wiki/Hexagonal_antiprism" title="Hexagonal antiprism">hexagonal antiprism</a>, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>6d</sub></a> symmetry, order 24</li> <li><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">Triakis tetrahedron</a> – 12 isosceles triangles, dual of <a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a>, <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">T<sub>d</sub></a> symmetry, order 24</li></ul></li> <li>Other less regular faced: <ul><li>Hendecagonal <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a> – 11 isosceles triangles and 1 regular <a href="/wiki/Hendecagon" title="Hendecagon">hendecagon</a>, <a href="/wiki/Cyclic_symmetry" class="mw-redirect" title="Cyclic symmetry">C<sub>11v</sub></a>, order 11</li> <li><a href="/wiki/Trapezo-rhombic_dodecahedron" title="Trapezo-rhombic dodecahedron">Trapezo-rhombic dodecahedron</a> – 6 rhombi, 6 <a href="/wiki/Trapezoid" title="Trapezoid">trapezoids</a> – dual of <a href="/wiki/Triangular_orthobicupola" title="Triangular orthobicupola">triangular orthobicupola</a>, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>3h</sub></a> symmetry, order 12</li> <li><a href="/wiki/Rhombo-hexagonal_dodecahedron" class="mw-redirect" title="Rhombo-hexagonal dodecahedron">Rhombo-hexagonal dodecahedron</a> or <i>elongated Dodecahedron</i> – 8 rhombi and 4 equilateral <a href="/wiki/Hexagon" title="Hexagon">hexagons</a>, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>4h</sub></a> symmetry, order 16</li> <li><a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">Truncated pentagonal trapezohedron</a>, <a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">D<sub>5d</sub></a>, order 20, topologically equivalent to regular dodecahedron</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Practical_usage">Practical usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=13" title="Edit section: Practical usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Armand_Spitz" title="Armand Spitz">Armand Spitz</a> used a dodecahedron as the "globe" equivalent for his <a href="/wiki/Planetarium_projector" title="Planetarium projector">Digital Dome planetarium projector</a>,<sup id="cite_ref-ley196502_10-0" class="reference"><a href="#cite_note-ley196502-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> based upon a suggestion from <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>. </p><p>Regular dodecahedrons are sometimes used as dice, when they are known as d12s, especially in games such as <a href="/wiki/Dungeons_and_Dragons" class="mw-redirect" title="Dungeons and Dragons">Dungeons and Dragons</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/120-cell" title="120-cell">120-cell</a> – a <a href="/wiki/Convex_regular_4-polytope" class="mw-redirect" title="Convex regular 4-polytope">regular polychoron</a> (4D polytope) whose surface consists of 120 dodecahedral cells</li> <li><em><a href="/wiki/Braarudosphaera_bigelowii" title="Braarudosphaera bigelowii">Braarudosphaera bigelowii</a></em> – a dodecahedron shaped <a href="/wiki/Coccolithophore" title="Coccolithophore">coccolithophore</a> (a <a href="/wiki/Unicellular_organism" title="Unicellular organism">unicellular</a> <a href="/wiki/Phytoplankton" title="Phytoplankton">phytoplankton</a> <a href="/wiki/Algae" title="Algae">algae</a>)</li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">Pentakis dodecahedron</a></li> <li><a href="/wiki/Roman_dodecahedron" title="Roman dodecahedron">Roman dodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">Snub dodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">Truncated dodecahedron</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dodecahedron&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:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span 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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 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Retrieved on 2016-12-02.</span> </li> <li id="cite_note-ley196502-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-ley196502_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLey1965" class="citation magazine cs1">Ley, Willy (February 1965). <a rel="nofollow" class="external text" href="https://archive.org/stream/Galaxy_v23n02_1964-12#page/n93/mode/2up">"Forerunners of the Planetarium"</a>. For Your Information. <i>Galaxy Science Fiction</i>. pp. 87–98.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Galaxy+Science+Fiction&rft.atitle=Forerunners+of+the+Planetarium&rft.pages=87-98&rft.date=1965-02&rft.aulast=Ley&rft.aufirst=Willy&rft_id=https%3A%2F%2Farchive.org%2Fstream%2FGalaxy_v23n02_1964-12%23page%2Fn93%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodecahedron" class="Z3988"></span></span> </li> </ol></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=Dodecahedron&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"><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="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:Polyhedra_with_12_faces" class="extiw" title="commons:Category:Polyhedra with 12 faces">Polyhedra with 12 faces</a></span>.</div></div> </div> <ul><li><i>Plato's Fourth Solid and the "Pyritohedron"</i>, by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226 <a rel="nofollow" class="external autonumber" href="https://www.jstor.org/pss/3619718">[1]</a></li> <li><a rel="nofollow" class="external text" href="http://bulatov.org/polyhedra/dodeca270/index.html">Stellation of Pyritohedron</a> VRML models and animations of Pyritohedron and its <a href="/wiki/Stellation" title="Stellation">stellations</a></li> <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 o3o5x – doe"</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+o3o5x+%E2%80%93+doe&rft.aulast=Klitzing&rft.aufirst=Richard&rft_id=https%3A%2F%2Fbendwavy.org%2Fklitzing%2Fdimensions%2Fpolyhedra.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodecahedron" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1bk9bWiCSjJz6LpNRYDsAu8YDBWnSMrt0ydjpIfF8jmyc682nzINN9xaGayOA9FBx396IIYMhulg2mGXcK0mAk5Rmo8qm9ut0kE1qP&name=Dodecahedron#applet">Editable printable net of a dodecahedron with interactive 3D view</a></li> <li><a rel="nofollow" class="external text" href="http://www.mathconsult.ch/showroom/unipoly/">The Uniform Polyhedra</a></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="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="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.bodurov.com/VectorVisualizer/?vectors=-0.94/-2.885/-3.975/-1.52/-4.67/-0.94v-3.035/0/-3.975/-4.91/0/-0.94v3.975/-2.885/-0.94/1.52/-4.67/0.94v1.52/-4.67/0.94/-1.52/-4.67/-0.94v0.94/-2.885/3.975/1.52/-4.67/0.94v-3.975/-2.885/0.94/-1.52/-4.67/-0.94v-3.975/-2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-4.91/0/-0.94v-3.975/2.885/0.94/-1.52/4.67/-0.94v-2.455/1.785/3.975/-3.975/2.885/0.94v-2.455/-1.785/3.975/-3.975/-2.885/0.94v-1.52/4.67/-0.94/-0.94/2.885/-3.975v4.91/0/0.94/3.975/-2.885/-0.94v3.975/2.885/-0.94/2.455/1.785/-3.975v2.455/-1.785/-3.975/3.975/-2.885/-0.94v1.52/4.67/0.94/-1.52/4.67/-0.94v3.035/0/3.975/0.94/2.885/3.975v0.94/2.885/3.975/-2.455/1.785/3.975v-2.455/1.785/3.975/-2.455/-1.785/3.975v-2.455/-1.785/3.975/0.94/-2.885/3.975v0.94/-2.885/3.975/3.035/0/3.975v2.455/1.785/-3.975/-0.94/2.885/-3.975v-0.94/2.885/-3.975/-3.035/0/-3.975v-3.035/0/-3.975/-0.94/-2.885/-3.975v-0.94/-2.885/-3.975/2.455/-1.785/-3.975v2.455/-1.785/-3.975/2.455/1.785/-3.97v3.035/0/3.975/4.91/0/0.94v4.91/0/0.94/3.975/2.885/-0.94v3.975/2.885/-0.94/1.52/4.67/0.94v1.52/4.67/0.94/0.94/2.885/3.975">Dodecahedron 3D Visualization</a></li> <li><a rel="nofollow" class="external text" href="http://www.software3d.com/Stella.php">Stella: Polyhedron Navigator</a>: Software used to create some of the images on this page.</li> <li><a rel="nofollow" class="external text" href="http://video.fc2.com/content/20141015mMG9QR5R">How to make a dodecahedron from a Styrofoam cube</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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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 class="mw-selflink selflink">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 href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a><br /><span class="nobold">(<a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li> <li><a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li> <li><a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a></li> <li><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li> <li><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li> <li><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li> <li><a href="/wiki/Snub_cube" title="Snub cube">snub cube</a></li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li> <li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li> <li><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li> <li><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a><br /><span class="nobold">(duals of Archimedean)</span></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li> <li><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li> <li><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li> <li><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li> <li><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li> <li><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li> <li><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li> <li><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Dihedron" title="Dihedron">dihedron</a></i></li> <li><i><a href="/wiki/Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a></li> <li><a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral others</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></li> <li><a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedra</a></li> <li><a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramid</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Bicupola_(geometry)" class="mw-redirect" title="Bicupola (geometry)">bicupola</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Bifrustum" title="Bifrustum">bifrustum</a></li> <li><a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotunda</a></li> <li><a href="/wiki/Birotunda" title="Birotunda">birotunda</a></li> <li><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a></li> <li><a href="/wiki/Scutoid" title="Scutoid">scutoid</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Degenerate polyhedra are in <i>italics</i>.</div></td></tr></tbody></table></div> <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 href="/wiki/Regular_icosahedron" title="Regular icosahedron">Icosahedron</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polychoron" class="mw-redirect" title="Uniform polychoron">Uniform polychoron</a> </th> <td><a href="/wiki/5-cell" title="5-cell">Pentachoron</a> </td> <td><a href="/wiki/16-cell" title="16-cell">16-cell</a> • <a href="/wiki/Tesseract" title="Tesseract">Tesseract</a> </td> <td><a href="/wiki/16-cell" title="16-cell">Demitesseract</a> </td> <td style="background:#e0f0e0;"><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td> <td><a href="/wiki/120-cell" title="120-cell">120-cell</a> • <a href="/wiki/600-cell" title="600-cell">600-cell</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_5-polytope" title="Uniform 5-polytope">Uniform 5-polytope</a> </th> <td style="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> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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