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class="vector-toc-list"> <li id="toc-Measurement_and_other_metric_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement_and_other_metric_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Measurement and other metric properties</span> </div> </a> <ul id="toc-Measurement_and_other_metric_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_spheres" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_spheres"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Relation to the spheres</span> </div> </a> <ul id="toc-Relation_to_the_spheres-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Symmetry</span> </div> </a> <ul id="toc-Symmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Construction</span> </div> </a> <ul id="toc-Construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Representation</span> </div> </a> <button aria-controls="toc-Representation-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 Representation subsection</span> </button> <ul id="toc-Representation-sublist" class="vector-toc-list"> <li id="toc-As_a_graph" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_graph"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>As a graph</span> </div> </a> <ul id="toc-As_a_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_orthogonal_projection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_orthogonal_projection"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>In orthogonal projection</span> </div> </a> <ul id="toc-In_orthogonal_projection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_configuration_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_configuration_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>As a configuration matrix</span> </div> </a> <ul id="toc-As_a_configuration_matrix-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_figures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_figures"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Related figures</span> </div> </a> <button aria-controls="toc-Related_figures-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Related figures subsection</span> </button> <ul id="toc-Related_figures-sublist" class="vector-toc-list"> <li id="toc-Construction_of_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_of_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Construction of polyhedra</span> </div> </a> <ul id="toc-Construction_of_polyhedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Space-filling_and_honeycombs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Space-filling_and_honeycombs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Space-filling and honeycombs</span> </div> </a> <ul id="toc-Space-filling_and_honeycombs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Miscellaneous"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Miscellaneous</span> </div> </a> <ul id="toc-Miscellaneous-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Cube</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 105 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-105" 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">105 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/Kubus" title="Kubus – Afrikaans" lang="af" hreflang="af" data-title="Kubus" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%AD%E1%8B%A9%E1%89%A5" title="ክዩብ – Amharic" lang="am" hreflang="am" data-title="ክዩብ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%83%D8%B9%D8%A8" title="مكعب – Arabic" lang="ar" hreflang="ar" data-title="مكعب" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Cubo" title="Cubo – Aragonese" lang="an" hreflang="an" data-title="Cubo" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D4%BD%D5%B8%D6%80%D5%A1%D5%B6%D5%A1%D6%80%D5%A4" title="Խորանարդ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Խորանարդ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Hexaedru" title="Hexaedru – Asturian" lang="ast" hreflang="ast" data-title="Hexaedru" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kub" title="Kub – Azerbaijani" lang="az" hreflang="az" data-title="Kub" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%DB%86%D8%A8" title="کۆب – South Azerbaijani" lang="azb" hreflang="azb" data-title="کۆب" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Chi%C3%A0%E2%81%BF_lio%CC%8Dk-bi%C4%81n-th%C3%A9" title="Chiàⁿ lio̍k-biān-thé – Minnan" lang="nan" hreflang="nan" data-title="Chiàⁿ lio̍k-biān-thé" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D1%83%D0%B1" title="Куб – Bashkir" lang="ba" hreflang="ba" data-title="Куб" 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%9A%D1%83%D0%B1" title="Куб – Belarusian" lang="be" hreflang="be" data-title="Куб" 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%9A%D1%83%D0%B1" title="Куб – Bulgarian" lang="bg" hreflang="bg" data-title="Куб" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kocka" title="Kocka – Bosnian" lang="bs" hreflang="bs" data-title="Kocka" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cub" title="Cub – Catalan" lang="ca" hreflang="ca" data-title="Cub" 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%9A%D1%83%D0%B1" title="Куб – Chuvash" lang="cv" hreflang="cv" data-title="Куб" 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/Krychle" title="Krychle – Czech" lang="cs" hreflang="cs" data-title="Krychle" 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/Ciwb" title="Ciwb – Welsh" lang="cy" hreflang="cy" data-title="Ciwb" 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/Terning" title="Terning – Danish" lang="da" hreflang="da" data-title="Terning" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D9%83%D8%B9%D8%A8" title="مكعب – Moroccan Arabic" lang="ary" hreflang="ary" data-title="مكعب" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Gu%C4%91ahas" title="Guđahas – Northern Sami" lang="se" hreflang="se" data-title="Guđahas" data-language-autonym="Davvisámegiella" data-language-local-name="Northern Sami" class="interlanguage-link-target"><span>Davvisámegiella</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/W%C3%BCrfel_(Geometrie)" title="Würfel (Geometrie) – German" lang="de" hreflang="de" data-title="Würfel (Geometrie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kuup" title="Kuup – Estonian" lang="et" hreflang="et" data-title="Kuup" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8D%CE%B2%CE%BF%CF%82" title="Κύβος – Greek" lang="el" hreflang="el" data-title="Κύβος" 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/Cubo" title="Cubo – Spanish" lang="es" hreflang="es" data-title="Cubo" 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/Kubo_(geometrio)" title="Kubo (geometrio) – Esperanto" lang="eo" hreflang="eo" data-title="Kubo (geometrio)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kubo" title="Kubo – Basque" lang="eu" hreflang="eu" data-title="Kubo" 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/%D9%85%DA%A9%D8%B9%D8%A8" title="مکعب – Persian" lang="fa" hreflang="fa" data-title="مکعب" 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/Cube" title="Cube – French" lang="fr" hreflang="fr" data-title="Cube" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ci%C3%BAb" title="Ciúb – Irish" lang="ga" hreflang="ga" data-title="Ciúb" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cubo" title="Cubo – Galician" lang="gl" hreflang="gl" data-title="Cubo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%AB%8B%E6%96%B9%E9%AB%94" title="立方體 – Gan" lang="gan" hreflang="gan" data-title="立方體" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%AE%E0%AA%98%E0%AA%A8" title="સમઘન – Gujarati" lang="gu" hreflang="gu" data-title="સમઘન" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%9C%A1%EB%A9%B4%EC%B2%B4" title="정육면체 – Korean" lang="ko" hreflang="ko" data-title="정육면체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BD%D5%B8%D6%80%D5%A1%D5%B6%D5%A1%D6%80%D5%A4" title="Խորանարդ – Armenian" lang="hy" hreflang="hy" data-title="Խորանարդ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%98%E0%A4%A8_(%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF)" title="घन (ज्यामिति) – Hindi" lang="hi" hreflang="hi" data-title="घन (ज्यामिति)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kocka" title="Kocka – Croatian" lang="hr" hreflang="hr" data-title="Kocka" 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/Kubo" title="Kubo – Ido" lang="io" hreflang="io" data-title="Kubo" 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/Kubus" title="Kubus – Indonesian" lang="id" hreflang="id" data-title="Kubus" 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-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/IGcikiva" title="IGcikiva – Zulu" lang="zu" hreflang="zu" data-title="IGcikiva" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Cubo" title="Cubo – Italian" lang="it" hreflang="it" data-title="Cubo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%91%D7%99%D7%99%D7%94" title="קובייה – Hebrew" lang="he" hreflang="he" data-title="קובייה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Kubuk" title="Kubuk – Javanese" lang="jv" hreflang="jv" data-title="Kubuk" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%98%E0%B2%A8_(%E0%B2%86%E0%B2%95%E0%B3%83%E0%B2%A4%E0%B2%BF)" title="ಘನ (ಆಕೃತಿ) – Kannada" lang="kn" hreflang="kn" data-title="ಘನ (ಆಕೃತಿ)" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%A3%E1%83%91%E1%83%98" title="კუბი – Georgian" lang="ka" hreflang="ka" data-title="კუბი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D1%83%D0%B1" title="Куб – Kazakh" lang="kk" hreflang="kk" data-title="Куб" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mchemraba" title="Mchemraba – Swahili" lang="sw" hreflang="sw" data-title="Mchemraba" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D1%83%D0%B1" title="Куб – Kyrgyz" lang="ky" hreflang="ky" data-title="Куб" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%A5%E0%BA%B9%E0%BA%81%E0%BA%9A%E0%BA%B2%E0%BA%94" title="ລູກບາດ – Lao" lang="lo" hreflang="lo" data-title="ລູກບາດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Cubus" title="Cubus – Latin" lang="la" hreflang="la" data-title="Cubus" 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/Kubs" title="Kubs – Latvian" lang="lv" hreflang="lv" data-title="Kubs" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Wierfel" title="Wierfel – Luxembourgish" lang="lb" hreflang="lb" data-title="Wierfel" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kubas" title="Kubas – Lithuanian" lang="lt" hreflang="lt" data-title="Kubas" 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/Kocka" title="Kocka – Hungarian" lang="hu" hreflang="hu" data-title="Kocka" 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%9A%D0%BE%D1%86%D0%BA%D0%B0" title="Коцка – Macedonian" lang="mk" hreflang="mk" data-title="Коцка" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Goba_(je%C3%B4metria)" title="Goba (jeômetria) – Malagasy" lang="mg" hreflang="mg" data-title="Goba (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%AE%E0%B4%9A%E0%B4%A4%E0%B5%81%E0%B4%B0%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B4%9F%E0%B5%8D%E0%B4%9F" title="സമചതുരക്കട്ട – Malayalam" lang="ml" hreflang="ml" data-title="സമചതുരക്കട്ട" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%98%E0%A4%A8_(%E0%A4%AD%E0%A5%82%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80)" title="घन (भूमिती) – Marathi" lang="mr" hreflang="mr" data-title="घन (भूमिती)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kubus" title="Kubus – Malay" lang="ms" hreflang="ms" data-title="Kubus" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%9A%D1%83%D0%B1%D1%81%D1%8C" title="Кубсь – Moksha" lang="mdf" hreflang="mdf" data-title="Кубсь" data-language-autonym="Мокшень" data-language-local-name="Moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kubus_(ruimtelijke_figuur)" title="Kubus (ruimtelijke figuur) – Dutch" lang="nl" hreflang="nl" data-title="Kubus (ruimtelijke figuur)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E5%85%AD%E9%9D%A2%E4%BD%93" title="正六面体 – Japanese" lang="ja" hreflang="ja" data-title="正六面体" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Tiarling_(Geometrii)" title="Tiarling (Geometrii) – Northern Frisian" lang="frr" hreflang="frr" data-title="Tiarling (Geometrii)" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kube" title="Kube – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kube" 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/Kube" title="Kube – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kube" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Cube" title="Cube – Occitan" lang="oc" hreflang="oc" data-title="Cube" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9A%D1%83%D0%B1" title="Куб – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Куб" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Garjii" title="Garjii – Oromo" lang="om" hreflang="om" data-title="Garjii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kub" title="Kub – Uzbek" lang="uz" hreflang="uz" data-title="Kub" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%98%E0%A8%A3_(%E0%A8%96%E0%A9%87%E0%A8%A4%E0%A8%B0%E0%A8%AE%E0%A8%BF%E0%A8%A4%E0%A9%80)" title="ਘਣ (ਖੇਤਰਮਿਤੀ) – Punjabi" lang="pa" hreflang="pa" data-title="ਘਣ (ਖੇਤਰਮਿਤੀ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DA%A9%D8%B9%D8%A8" title="مکعب – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مکعب" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF%D8%B1%DB%90%D9%8A%DA%81_(%D9%85%DB%90%DA%86%D9%BE%D9%88%D9%87%D9%86%D9%87)" title="درېيځ (مېچپوهنه) – Pashto" lang="ps" hreflang="ps" data-title="درېيځ (مېچپوهنه)" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kyuub" title="Kyuub – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Kyuub" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Cubo" title="Cubo – Piedmontese" lang="pms" hreflang="pms" data-title="Cubo" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/W%C3%B6rpel_(Geometrie)" title="Wörpel (Geometrie) – Low German" lang="nds" hreflang="nds" data-title="Wörpel (Geometrie)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sze%C5%9Bcian_(geometria)" title="Sześcian (geometria) – Polish" lang="pl" hreflang="pl" data-title="Sześcian (geometria)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Cubo" title="Cubo – Portuguese" lang="pt" hreflang="pt" data-title="Cubo" 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/Cub" title="Cub – Romanian" lang="ro" hreflang="ro" data-title="Cub" 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/Mach%27ina" title="Mach'ina – Quechua" lang="qu" hreflang="qu" data-title="Mach'ina" 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%9A%D1%83%D0%B1" title="Куб – Russian" lang="ru" hreflang="ru" data-title="Куб" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D1%83%D0%B1" title="Куб – Yakut" lang="sah" hreflang="sah" data-title="Куб" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Cube" title="Cube – Scots" lang="sco" hreflang="sco" data-title="Cube" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Kubi_(gjeometri)" title="Kubi (gjeometri) – Albanian" lang="sq" hreflang="sq" data-title="Kubi (gjeometri)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Cubbu" title="Cubbu – Sicilian" lang="scn" hreflang="scn" data-title="Cubbu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9D%E0%B6%B1%E0%B6%9A%E0%B6%BA" title="ඝනකය – Sinhala" lang="si" hreflang="si" data-title="ඝනකය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Cube" title="Cube – Simple English" lang="en-simple" hreflang="en-simple" data-title="Cube" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kocka" title="Kocka – Slovak" lang="sk" hreflang="sk" data-title="Kocka" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kocka" title="Kocka – Slovenian" lang="sl" hreflang="sl" data-title="Kocka" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Sedjibeke_(shaxan)" title="Sedjibeke (shaxan) – Somali" lang="so" hreflang="so" data-title="Sedjibeke (shaxan)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B4%DB%95%D8%B4%D9%BE%D8%A7%DA%B5%D9%88%D9%88" title="شەشپاڵوو – Central Kurdish" lang="ckb" hreflang="ckb" data-title="شەشپاڵوو" 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%9A%D0%BE%D1%86%D0%BA%D0%B0" title="Коцка – Serbian" lang="sr" hreflang="sr" data-title="Коцка" 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/Kocka" title="Kocka – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kocka" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Kubes" title="Kubes – Sundanese" lang="su" hreflang="su" data-title="Kubes" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kuutio" title="Kuutio – Finnish" lang="fi" hreflang="fi" data-title="Kuutio" 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/Kub" title="Kub – Swedish" lang="sv" hreflang="sv" data-title="Kub" 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%95%E0%AE%A9%E0%AE%9A%E0%AE%A4%E0%AF%81%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="கனசதுரம் – Tamil" lang="ta" hreflang="ta" data-title="கனசதுரம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8D%E0%B0%AF%E0%B1%82%E0%B0%AC%E0%B1%8D" title="క్యూబ్ – Telugu" lang="te" hreflang="te" data-title="క్యూబ్" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</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%A5%E0%B8%B9%E0%B8%81%E0%B8%9A%E0%B8%B2%E0%B8%A8%E0%B8%81%E0%B9%8C" title="ทรงลูกบาศก์ – Thai" lang="th" hreflang="th" data-title="ทรงลูกบาศก์" 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/K%C3%BCp" title="Küp – Turkish" lang="tr" hreflang="tr" 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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">Solid object with six equal square 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">For other uses, see <a href="/wiki/Cube_(disambiguation)" class="mw-disambig" title="Cube (disambiguation)">Cube (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Cube</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Cube-h.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Cube-h.svg/220px-Cube-h.svg.png" decoding="async" width="220" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Cube-h.svg/330px-Cube-h.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Cube-h.svg/440px-Cube-h.svg.png 2x" data-file-width="417" data-file-height="453" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Hanner_polytope" title="Hanner polytope">Hanner polytope</a>,<br /><a href="/wiki/Orthogonal_polyhedron" title="Orthogonal polyhedron">orthogonal polyhedron</a>,<br /><a href="/wiki/Parallelohedron" title="Parallelohedron">parallelohedron</a>,<br /><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>,<br /><a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedron</a>,<br /><a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular polyhedron</a>,<br /><a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Face_(geometry)" title="Face (geometry)">Faces</a></th><td class="infobox-data">6</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></th><td class="infobox-data">12</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></th><td class="infobox-data">8</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_configuration" title="Vertex configuration">Vertex configuration</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8\times (4^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>×</mo> <mo stretchy="false">(</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8\times (4^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7175a5b09172b2b75bbf8b5d682c2d3880d520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.029ex; height:3.176ex;" alt="{\displaystyle 8\times (4^{3})}" /></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{4,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{4,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f119d7ae6f9f443d28af07cc9a03657bea37dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{4,3\}}" /></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">Symmetry group</a></th><td class="infobox-data"><a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} _{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} _{\mathrm {h} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c0a7669167dbe00f64525845a9c7bff1b39ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.954ex; height:2.509ex;" alt="{\displaystyle \mathrm {O} _{\mathrm {h} }}" /></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dihedral_angle" title="Dihedral angle">Dihedral angle</a> (<a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>)</th><td class="infobox-data">90°</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></th><td class="infobox-data"><a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a></td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="/wiki/Convex_set" title="Convex set">convex</a>,<br /><a href="/wiki/Edge-transitive" class="mw-redirect" title="Edge-transitive">edge-transitive</a>,<br /><a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>,<br /><a href="/wiki/Non-composite_polyhedron" class="mw-redirect" title="Non-composite polyhedron">non-composite</a>,<br /><a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> faces,<br /><a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a></td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>cube</b> or <b>regular hexahedron</b> is a <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> solid object bounded by six congruent <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> faces, a type of <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a>. It has twelve congruent edges and eight vertices. It is a type of <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a>, with pairs of parallel opposite faces, and more specifically a <a href="/wiki/Rhombohedron" title="Rhombohedron">rhombohedron</a>, with congruent edges, and a <a href="/wiki/Rectangular_cuboid" title="Rectangular cuboid">rectangular cuboid</a>, with <a href="/wiki/Right_angle" title="Right angle">right angles</a> between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>, <a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular polyhedron</a>, <a href="/wiki/Parallelohedron" title="Parallelohedron">parallelohedron</a>, <a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a>, and <a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedron</a>. The <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> of a cube is the <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>. </p><p>The cube can be represented in many ways, one of which is the graph known as the <b>cubical graph</b>. It can be constructed by using the <a href="/wiki/Cartesian_product_of_graphs" title="Cartesian product of graphs">Cartesian product of graphs</a>. The cube is the three-dimensional <a href="/wiki/Hypercube" title="Hypercube">hypercube</a>, a family of <a href="/wiki/Polytope" title="Polytope">polytopes</a> also including the two-dimensional square and four-dimensional <a href="/wiki/Tesseract" title="Tesseract">tesseract</a>. A cube with <a href="/wiki/1" title="1">unit</a> side length is the canonical unit of <a href="/wiki/Volume" title="Volume">volume</a> in three-dimensional space, relative to which other solid objects are measured. Other related figures involve the construction of polyhedra, <a href="/wiki/Space-filling_polyhedron" title="Space-filling polyhedron">space-filling</a> and <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">honeycombs</a>, <a href="/wiki/Polycube" title="Polycube">polycubes</a>, as well as cube in compounds, spherical, and topological space. </p><p>The cube was discovered in antiquity, associated with the nature of <a href="/wiki/Earth_(classical_element)" title="Earth (classical element)">earth</a> by <a href="/wiki/Plato" title="Plato">Plato</a>, the founder of Platonic solid. It was used as a part of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>, proposed by <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a>. It can be derived differently to create more polyhedrons, and it has applications to construct a new <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> by attaching others. Other applications include popular culture of toys and games, arts, optical illusions, architectural buildings, as well as the natural science and technology. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span class="mw-3d-wrapper" data-label="3D"><a href="/wiki/File:Hexahedron.stl" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Hexahedron.stl/220px-Hexahedron.stl.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Hexahedron.stl/330px-Hexahedron.stl.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Hexahedron.stl/440px-Hexahedron.stl.png 2x" data-file-width="5120" data-file-height="2880" /></a></span><figcaption>3D model of a cube</figcaption></figure> <p>A cube is a special case of <a href="/wiki/Rectangular_cuboid" title="Rectangular cuboid">rectangular cuboid</a> in which the edges are equal in length.<sup id="cite_ref-mk_1-0" class="reference"><a href="#cite_note-mk-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Like other cuboids, every face of a cube has four vertices, each of which connects with three congruent lines. These edges form <a href="/wiki/Square" title="Square">square</a> faces, making the <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> of a cube between every two adjacent squares being the <a href="/wiki/Interior_angle" class="mw-redirect" title="Interior angle">interior angle</a> of a square, 90°. Hence, the cube has six faces, twelve edges, and eight vertices. <sup id="cite_ref-johnson_2-0" class="reference"><a href="#cite_note-johnson-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Because of such properties, it is categorized as one of the five <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>, a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> in which all the <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygons</a> are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> and the same number of faces meet at each vertex.<sup id="cite_ref-hs_3-0" class="reference"><a href="#cite_note-hs-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Every three square faces surrounding a vertex is <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> each other, so the cube is classified as <a href="/wiki/Orthogonal_polyhedron" title="Orthogonal polyhedron">orthogonal polyhedron</a>.<sup id="cite_ref-jessen_4-0" class="reference"><a href="#cite_note-jessen-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The cube may also be considered as the <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> in which all of its edges are equal edges.<sup id="cite_ref-calter_5-0" class="reference"><a href="#cite_note-calter-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Measurement_and_other_metric_properties">Measurement and other metric properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=2" title="Edit section: Measurement and other metric properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cube_diagonals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/130px-Cube_diagonals.svg.png" decoding="async" width="130" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/195px-Cube_diagonals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/260px-Cube_diagonals.svg.png 2x" data-file-width="275" data-file-height="265" /></a><figcaption>A face diagonal in red and space diagonal in blue</figcaption></figure> <p>Given a cube with edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span>. The <a href="/wiki/Face_diagonal" title="Face diagonal">face diagonal</a> of a cube is the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> of a square <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f5c1e0c6480a9d660cc0e67063dfc3b872654a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.328ex; height:3.009ex;" alt="{\displaystyle a{\sqrt {2}}}" /></span>, and the <a href="/wiki/Space_diagonal" title="Space diagonal">space diagonal</a> of a cube is a line connecting two vertices that is not in the same face, formulated as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c265627f35e18499654722022d0dee669febfe19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.328ex; height:2.843ex;" alt="{\displaystyle a{\sqrt {3}}}" /></span>. Both formulas can be determined by using <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The surface area of a cube <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /></span> is six times the area of a square:<sup id="cite_ref-khattar_6-0" class="reference"><a href="#cite_note-khattar-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=6a^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>6</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=6a^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29b63c94a8387e8174bc6bb41042dc2cfa9ec78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.935ex; height:2.676ex;" alt="{\displaystyle A=6a^{2}.}" /></span> The volume of a cuboid is the product of its length, width, and height. Because all the edges of a cube are equal in length, the formula for the volume of a cube as the third power of its side length, leading to the use of the term <i><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">cubic</a></i> to mean raising any number to the third power:<sup id="cite_ref-thomson_7-0" class="reference"><a href="#cite_note-thomson-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-khattar_6-1" class="reference"><a href="#cite_note-khattar-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=a^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=a^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a4c0cc0c239e1b1b30998a0b475a51d38dd97f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.817ex; height:2.676ex;" alt="{\displaystyle V=a^{3}.}" /></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Prince_Ruperts_cube.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Prince_Ruperts_cube.png/130px-Prince_Ruperts_cube.png" decoding="async" width="130" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Prince_Ruperts_cube.png/195px-Prince_Ruperts_cube.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Prince_Ruperts_cube.png/260px-Prince_Ruperts_cube.png 2x" data-file-width="465" data-file-height="522" /></a><figcaption>The <a href="/wiki/Prince_Rupert%27s_cube" title="Prince Rupert's cube">Prince Rupert's cube</a></figcaption></figure> <p>One special case is the <a href="/wiki/Unit_cube" title="Unit cube">unit cube</a>, so named for measuring a single <a href="/wiki/Unit_of_length" title="Unit of length">unit of length</a> along each edge. It follows that each face is a <a href="/wiki/Unit_square" title="Unit square">unit square</a> and that the entire figure has a volume of 1 cubic unit.<sup id="cite_ref-ball_8-0" class="reference"><a href="#cite_note-ball-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hr-w_9-0" class="reference"><a href="#cite_note-hr-w-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Prince_Rupert%27s_cube" title="Prince Rupert's cube">Prince Rupert's cube</a>, named after <a href="/wiki/Prince_Rupert_of_the_Rhine" title="Prince Rupert of the Rhine">Prince Rupert of the Rhine</a>, is the largest cube that can pass through a hole cut into the unit cube, despite having sides approximately 6% longer.<sup id="cite_ref-sriraman_10-0" class="reference"><a href="#cite_note-sriraman-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A polyhedron that can pass through a copy of itself of the same size or smaller is said to have the <a href="/wiki/Rupert_property" class="mw-redirect" title="Rupert property">Rupert property</a>.<sup id="cite_ref-jwy_11-0" class="reference"><a href="#cite_note-jwy-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> A geometric problem of <a href="/wiki/Doubling_the_cube" title="Doubling the cube">doubling the cube</a>—alternatively known as the <i>Delian problem</i>—requires the construction of a cube with a volume twice the original by using a <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a> solely. Ancient mathematicians could not solve this old problem until French mathematician <a href="/wiki/Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> in 1837 proved it was impossible.<sup id="cite_ref-lutzen_12-0" class="reference"><a href="#cite_note-lutzen-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_spheres">Relation to the spheres</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=3" title="Edit section: Relation to the spheres"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With edge length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span>, the <a href="/wiki/Inscribed_sphere" title="Inscribed sphere">inscribed sphere</a> of a cube is the sphere tangent to the faces of a cube at their centroids, with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c80af0f6bd462d58cdeef00e005c9ed9d8182e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.888ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}a}" /></span>. The <a href="/wiki/Midsphere" title="Midsphere">midsphere</a> of a cube is the sphere tangent to the edges of a cube, with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\sqrt {2}}{2}}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\sqrt {2}}{2}}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3fe7f0acc54574a31863824db814aaa73fb2553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.257ex; height:4.176ex;" alt="{\textstyle {\frac {\sqrt {2}}{2}}a}" /></span>. The <a href="/wiki/Circumscribed_sphere" title="Circumscribed sphere">circumscribed sphere</a> of a cube is the sphere tangent to the vertices of a cube, with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\sqrt {3}}{2}}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\sqrt {3}}{2}}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c587445ca61256c03872fd0a38b5047987e878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.257ex; height:4.176ex;" alt="{\textstyle {\frac {\sqrt {3}}{2}}a}" /></span>.<sup id="cite_ref-radii_13-0" class="reference"><a href="#cite_note-radii-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>For a cube whose circumscribed sphere has radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span>, and for a given point in its three-dimensional space with distances <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 d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}" /></span> from the cube's eight vertices, it is:<sup id="cite_ref-poo-sung_14-0" class="reference"><a href="#cite_note-poo-sung-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{4}+{\frac {16R^{4}}{9}}=\left({\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{2}+{\frac {2R^{2}}{3}}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </munderover> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>16</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </munderover> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{4}+{\frac {16R^{4}}{9}}=\left({\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{2}+{\frac {2R^{2}}{3}}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8204a09423743d2541f49dbcdc8a687977055abc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.758ex; height:8.009ex;" alt="{\displaystyle {\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{4}+{\frac {16R^{4}}{9}}=\left({\frac {1}{8}}\sum _{i=1}^{8}d_{i}^{2}+{\frac {2R^{2}}{3}}\right)^{2}.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=4" title="Edit section: Symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The cube has <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} _{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} _{\mathrm {h} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c0a7669167dbe00f64525845a9c7bff1b39ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.954ex; height:2.509ex;" alt="{\displaystyle \mathrm {O} _{\mathrm {h} }}" /></span>. It is composed of <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a>, a symmetry by cutting into two halves by a plane. There are nine reflection symmetries: the five are cut the cube from the midpoints of its edges, and the four are cut diagonally. It is also composed of <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a>, a symmetry by rotating it around the axis, from which the appearance is interchangeable. It has octahedral rotation symmetry <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 \mathrm {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d6d4173d32feed308e80dbaf00e1274f40702d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathrm {O} }" /></span>: three axes pass through the cube's opposite faces centroid, six through the cube's opposite edges midpoints, and four through the cube's opposite vertices; each of these axes is respectively four-fold rotational symmetry (0°, 90°, 180°, and 270°), two-fold rotational symmetry (0° and 180°), and three-fold rotational symmetry (0°, 120°, and 240°).<sup id="cite_ref-french_15-0" class="reference"><a href="#cite_note-french-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cromwell_16-0" class="reference"><a href="#cite_note-cromwell-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cp_17-0" class="reference"><a href="#cite_note-cp-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dual_Cube-Octahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/180px-Dual_Cube-Octahedron.svg.png" decoding="async" width="180" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/270px-Dual_Cube-Octahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/360px-Dual_Cube-Octahedron.svg.png 2x" data-file-width="744" data-file-height="749" /></a><figcaption>The dual polyhedron of a cube is the regular octahedron</figcaption></figure> <p>The <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> can be obtained from each of the polyhedron's vertices tangent to a plane by the process known as <a href="/wiki/Polar_reciprocation" class="mw-redirect" title="Polar reciprocation">polar reciprocation</a>.<sup id="cite_ref-cr_18-0" class="reference"><a href="#cite_note-cr-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> One property of dual polyhedrons generally is that the polyhedron and its dual share their <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">three-dimensional symmetry point group</a>. In this case, the dual polyhedron of a cube is the <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>, and both of these polyhedron has the same symmetry, the octahedral symmetry.<sup id="cite_ref-erickson_19-0" class="reference"><a href="#cite_note-erickson-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>The cube is <a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>, meaning its two squares are alike and can be mapped by rotation and reflection.<sup id="cite_ref-mclean_20-0" class="reference"><a href="#cite_note-mclean-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> It is <a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a>, meaning all of its vertices are equivalent and can be mapped <a href="/wiki/Isometry" title="Isometry">isometrically</a> under its symmetry.<sup id="cite_ref-grunbaum-1997_21-0" class="reference"><a href="#cite_note-grunbaum-1997-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> It is also <a href="/wiki/Edge-transitive" class="mw-redirect" title="Edge-transitive">edge-transitive</a>, meaning the same kind of faces surround each of its vertices in the same or reverse order, all two adjacent faces have the same <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a>. Therefore, the cube is <a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular polyhedron</a> because it requires those properties.<sup id="cite_ref-senechal_22-0" class="reference"><a href="#cite_note-senechal-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Each vertex is surrounded by three squares, so the cube is <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 4.4.4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4.4.4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4.4.4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca545ed1e391d4ad76a923393555f70660fb06c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.781ex; height:2.176ex;" alt="{\displaystyle 4.4.4}" /></span> by <a href="/wiki/Vertex_configuration" title="Vertex configuration">vertex configuration</a> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{4,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{4,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f119d7ae6f9f443d28af07cc9a03657bea37dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{4,3\}}" /></span> in <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a>.<sup id="cite_ref-wd_23-0" class="reference"><a href="#cite_note-wd-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/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 span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:352px;max-width:352px"><div class="trow"><div class="tsingle" style="width:123px;max-width:123px"><div class="thumbimage" style="height:125px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:One-red-dice-01.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/One-red-dice-01.jpg/121px-One-red-dice-01.jpg" decoding="async" width="121" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/One-red-dice-01.jpg/182px-One-red-dice-01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/One-red-dice-01.jpg/242px-One-red-dice-01.jpg 2x" data-file-width="336" data-file-height="348" /></a></span></div><div class="thumbcaption">A six-sided <a href="/wiki/Dice" title="Dice">dice</a></div></div><div class="tsingle" style="width:127px;max-width:127px"><div class="thumbimage" style="height:125px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Skewb.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Skewb.jpg/125px-Skewb.jpg" decoding="async" width="125" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Skewb.jpg/188px-Skewb.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Skewb.jpg/250px-Skewb.jpg 2x" data-file-width="1024" data-file-height="1024" /></a></span></div><div class="thumbcaption">A completed <a href="/wiki/Skewb" title="Skewb">Skewb</a></div></div><div class="tsingle" style="width:96px;max-width:96px"><div class="thumbimage" style="height:125px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:St_Marks_Place,_East_Village,_Downtown_New_York_City,_Recover_Reputation.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/St_Marks_Place%2C_East_Village%2C_Downtown_New_York_City%2C_Recover_Reputation.jpg/94px-St_Marks_Place%2C_East_Village%2C_Downtown_New_York_City%2C_Recover_Reputation.jpg" decoding="async" width="94" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/St_Marks_Place%2C_East_Village%2C_Downtown_New_York_City%2C_Recover_Reputation.jpg/141px-St_Marks_Place%2C_East_Village%2C_Downtown_New_York_City%2C_Recover_Reputation.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/St_Marks_Place%2C_East_Village%2C_Downtown_New_York_City%2C_Recover_Reputation.jpg/188px-St_Marks_Place%2C_East_Village%2C_Downtown_New_York_City%2C_Recover_Reputation.jpg 2x" data-file-width="2160" data-file-height="2880" /></a></span></div><div class="thumbcaption">A sculpture <a href="/wiki/Alamo_(sculpture)" title="Alamo (sculpture)"><i>Alamo</i></a></div></div></div></div></div> <p>Cubes have appeared in many popular cultures. In toys and games, <a href="/wiki/Dice" title="Dice">dice</a> are commonly found in a six-sided shape,<sup id="cite_ref-mclean_20-1" class="reference"><a href="#cite_note-mclean-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> puzzle toys such as <a href="/wiki/Rubik%27s_Cube" title="Rubik's Cube">Rubik's Cube</a> and <a href="/wiki/Skewb" title="Skewb">Skewb</a> are cube-shaped,<sup id="cite_ref-joyner_24-0" class="reference"><a href="#cite_note-joyner-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Sandbox_game" title="Sandbox game">sandbox video games</a> of cubic blocks with one example is <i><a href="/wiki/Minecraft" title="Minecraft">Minecraft</a></i>.<sup id="cite_ref-moore_25-0" class="reference"><a href="#cite_note-moore-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> In art, a 1967 outdoor sculpture <a href="/wiki/Alamo_(sculpture)" title="Alamo (sculpture)"><i>Alamo</i></a> is a cube rotated on its corner in which a pole is hidden inside,<sup id="cite_ref-rz_26-0" class="reference"><a href="#cite_note-rz-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Optical_illusions" class="mw-redirect" title="Optical illusions">optical illusions</a> such as the <a href="/wiki/Impossible_cube" title="Impossible cube">impossible cube</a> and <a href="/wiki/Necker_cube" title="Necker cube">Necker cube</a>,<sup id="cite_ref-barrow_27-0" class="reference"><a href="#cite_note-barrow-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and stacked cubes forming a three-dimensional cross is examples of both <a href="/wiki/Salvador_Dal%C3%AD" title="Salvador Dalí">Salvador Dalí</a>'s 1954 painting <i><a href="/wiki/Corpus_Hypercubus" class="mw-redirect" title="Corpus Hypercubus">Corpus Hypercubus</a></i> and <a href="/wiki/Robert_A._Heinlein" title="Robert A. Heinlein">Robert A. Heinlein</a>'s 1940 short story "<a href="/wiki/And_He_Built_a_Crooked_House" title="And He Built a Crooked House">And He Built a Crooked House</a>".<sup id="cite_ref-kemp_28-0" class="reference"><a href="#cite_note-kemp-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-fowler_29-0" class="reference"><a href="#cite_note-fowler-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> In architecture, the cube was applied in <a href="/wiki/Leon_Battista_Alberti" title="Leon Battista Alberti">Alberti</a>'s 1450 <i><a href="/wiki/De_re_aedificatoria" title="De re aedificatoria">De re aedificatoria</a></i> treatise on first <a href="/wiki/Renaissance_architecture" title="Renaissance architecture">Renaissance architecture</a>,<sup id="cite_ref-march_30-0" class="reference"><a href="#cite_note-march-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> and <i><a href="/wiki/Kubuswoningen" class="mw-redirect" title="Kubuswoningen">Kubuswoningen</a></i> is known for a set of cubical shaped houses in which its <a href="/wiki/Hexagon" title="Hexagon">hexagonal</a> space diagonal becomes the main floor.<sup id="cite_ref-an_31-0" class="reference"><a href="#cite_note-an-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:352px;max-width:352px"><div class="trow"><div class="tsingle" style="width:108px;max-width:108px"><div class="thumbimage" style="height:123px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Cubic.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/106px-Cubic.svg.png" decoding="async" width="106" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/159px-Cubic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Cubic.svg/212px-Cubic.svg.png 2x" data-file-width="109" data-file-height="127" /></a></span></div><div class="thumbcaption">Simple cubic crystal structure</div></div><div class="tsingle" style="width:119px;max-width:119px"><div class="thumbimage" style="height:123px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:2780M-pyrite1.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/2780M-pyrite1.jpg/117px-2780M-pyrite1.jpg" decoding="async" width="117" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/2780M-pyrite1.jpg/176px-2780M-pyrite1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/2780M-pyrite1.jpg/234px-2780M-pyrite1.jpg 2x" data-file-width="1938" data-file-height="2051" /></a></span></div><div class="thumbcaption"><a href="/wiki/Pyrite" title="Pyrite">Pyrite</a> cubic crystals</div></div><div class="tsingle" style="width:119px;max-width:119px"><div class="thumbimage" style="height:123px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Cubane_molecule_ball.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Cubane_molecule_ball.png/117px-Cubane_molecule_ball.png" decoding="async" width="117" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Cubane_molecule_ball.png/176px-Cubane_molecule_ball.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Cubane_molecule_ball.png/234px-Cubane_molecule_ball.png 2x" data-file-width="1889" data-file-height="2000" /></a></span></div><div class="thumbcaption"><a href="/wiki/Ball-and-stick_model" title="Ball-and-stick model">Ball-and-stick model</a> of cubane</div></div></div></div></div> <p>Cubes are also found in natural science and technology. It is applied to the <a href="/wiki/Unit_cell" title="Unit cell">unit cell</a> of a crystal known as a <a href="/wiki/Cubic_crystal_system" title="Cubic crystal system">cubic crystal system</a>.<sup id="cite_ref-tisza_32-0" class="reference"><a href="#cite_note-tisza-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pyrite" title="Pyrite">Pyrite</a> is an example of a <a href="/wiki/Mineral" title="Mineral">mineral</a> with a commonly cubic shape, although there are many varied shapes.<sup id="cite_ref-hoffmann_33-0" class="reference"><a href="#cite_note-hoffmann-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Cubane" title="Cubane">Cubane</a> is a synthetic <a href="/wiki/Hydrocarbon" title="Hydrocarbon">hydrocarbon</a> consisting of eight carbon <a href="/wiki/Atom" title="Atom">atoms</a> arranged at the corners of a cube, with one <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> atom attached to each carbon atom.<sup id="cite_ref-biegasiewicz_34-0" class="reference"><a href="#cite_note-biegasiewicz-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Several <a href="/wiki/Radiolarian" class="mw-redirect" title="Radiolarian">Radiolarians</a> were discovered by <a href="/wiki/Ernest_Haeckel" class="mw-redirect" title="Ernest Haeckel">Ernest Haeckel</a>, one of which was <i>Lithocubus geometricus</i> with a cubic shape.<sup id="cite_ref-haeckel_35-0" class="reference"><a href="#cite_note-haeckel-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Cubical grids are most commonly found in three-dimensional <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate systems</a>.<sup id="cite_ref-knstv_36-0" class="reference"><a href="#cite_note-knstv-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, <a href="/wiki/Marching_cubes" title="Marching cubes">an algorithm</a> divides the input volume into a discrete set of cubes known as the unit on <a href="/wiki/Isosurface" title="Isosurface">isosurface</a>,<sup id="cite_ref-cmsi_37-0" class="reference"><a href="#cite_note-cmsi-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> and the faces of a cube can be used for <a href="/wiki/Cube_mapping" title="Cube mapping">mapping a shape</a>.<sup id="cite_ref-greene_38-0" class="reference"><a href="#cite_note-greene-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> A historical attempt to unify three physics ideas of <a href="/wiki/Galilean_relativity" class="mw-redirect" title="Galilean relativity">relativity</a>, <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a>, and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> used the framework of a cube known as a <a href="/wiki/CGh_physics" title="CGh physics"><i>cGh</i> cube</a>.<sup id="cite_ref-padmanabhan_39-0" class="reference"><a href="#cite_note-padmanabhan-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Others are the spacecraft device <a href="/wiki/CubeSat" title="CubeSat">CubeSat</a>,<sup id="cite_ref-helvajian_40-0" class="reference"><a href="#cite_note-helvajian-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a> demonstration device <a href="/wiki/Leslie_cube" title="Leslie cube">Leslie cube</a>.<sup id="cite_ref-vm_41-0" class="reference"><a href="#cite_note-vm-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage" style="height:157px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Kepler_Hexahedron_Earth.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Kepler_Hexahedron_Earth.jpg/150px-Kepler_Hexahedron_Earth.jpg" decoding="async" width="150" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Kepler_Hexahedron_Earth.jpg/225px-Kepler_Hexahedron_Earth.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/5/57/Kepler_Hexahedron_Earth.jpg 2x" data-file-width="290" data-file-height="304" /></a></span></div><div class="thumbcaption">Sketch of a cube by Johannes Kepler</div></div><div class="tsingle" style="width:136px;max-width:136px"><div class="thumbimage" style="height:157px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Mysterium_Cosmographicum_solar_system_model.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Mysterium_Cosmographicum_solar_system_model.jpg/134px-Mysterium_Cosmographicum_solar_system_model.jpg" decoding="async" width="134" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Mysterium_Cosmographicum_solar_system_model.jpg/201px-Mysterium_Cosmographicum_solar_system_model.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Mysterium_Cosmographicum_solar_system_model.jpg/268px-Mysterium_Cosmographicum_solar_system_model.jpg 2x" data-file-width="3252" data-file-height="3832" /></a></span></div><div class="thumbcaption"><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler's</a> Platonic solid model of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a></div></div></div></div></div> <p>The <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a> is a set of polyhedrons known since antiquity. It was named after <a href="/wiki/Plato" title="Plato">Plato</a> in his <a href="/wiki/Timaeus_(dialogue)" title="Timaeus (dialogue)"><i>Timaeus</i></a> dialogue, who attributed these solids to nature. One of them, the cube, represented the <a href="/wiki/Classical_element" title="Classical element">classical element</a> of <a href="/wiki/Earth_(classical_element)" title="Earth (classical element)">earth</a> because of its stability.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]_42-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a> defined the Platonic solids, including the cube, and using these solids with the problem involving to find the ratio of the circumscribed sphere's diameter to the edge length.<sup id="cite_ref-heath_43-0" class="reference"><a href="#cite_note-heath-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Following its attribution with nature by Plato, <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> in his <i><a href="/wiki/Harmonices_Mundi" class="mw-redirect" title="Harmonices Mundi">Harmonices Mundi</a></i> sketched each of the Platonic solids, one of them being a cube in which Kepler decorated a tree on it.<sup id="cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]_42-1" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage55_55]-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> In his <i><a href="/wiki/Mysterium_Cosmographicum" title="Mysterium Cosmographicum">Mysterium Cosmographicum</a></i>, Kepler also proposed the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>, <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a>, <a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">regular dodecahedron</a>, <a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedron</a>, and cube.<sup id="cite_ref-livio_44-0" class="reference"><a href="#cite_note-livio-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=6" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:The_11_cubic_nets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/The_11_cubic_nets.svg/220px-The_11_cubic_nets.svg.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/The_11_cubic_nets.svg/330px-The_11_cubic_nets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/The_11_cubic_nets.svg/440px-The_11_cubic_nets.svg.png 2x" data-file-width="300" data-file-height="140" /></a><figcaption>Nets of a cube</figcaption></figure> <p>An elementary way to construct is using its <a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">net</a>, an arrangement of edge-joining polygons, constructing a polyhedron by connecting along the edges of those polygons. Eleven nets for the cube are shown here.<sup id="cite_ref-jeon_45-0" class="reference"><a href="#cite_note-jeon-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, a cube may be constructed using the <a href="/wiki/Cartesian_coordinate_systems" class="mw-redirect" title="Cartesian coordinate systems">Cartesian coordinate systems</a>. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of the vertices are <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 (\pm 1,\pm 1,\pm 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pm 1,\pm 1,\pm 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6152764ebd90dc441dfb5e681af0e9c45c41067c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.789ex; height:2.843ex;" alt="{\displaystyle (\pm 1,\pm 1,\pm 1)}" /></span>.<sup id="cite_ref-smith_46-0" class="reference"><a href="#cite_note-smith-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Its interior consists of all points <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 (x_{0},x_{1},x_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},x_{1},x_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba969259566b79a2af66826e29ccf36732ff1f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.029ex; height:2.843ex;" alt="{\displaystyle (x_{0},x_{1},x_{2})}" /></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1<x_{i}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1<x_{i}<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73df5ce0b50971fd4aad2a51c1a0463fb8c05f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.459ex; height:2.509ex;" alt="{\displaystyle -1<x_{i}<1}" /></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span>. A cube's surface with center <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 (x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}" /></span> and edge length of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d325c24be7d760207674a169b078892bdd5cbc76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.392ex; height:2.176ex;" alt="{\displaystyle 2a}" /></span> is the <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of all points <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 (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf86b1adcc2f8ca8cadcea5041c795d416c26d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.382ex; height:2.843ex;" alt="{\displaystyle \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.}" /></span> </p><p>The cube is <a href="/wiki/Hanner_polytope" title="Hanner polytope">Hanner polytope</a>, because it can be constructed by using <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of three line segments. Its dual polyhedron, the regular octahedron, is constructed by <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a> of three line segments.<sup id="cite_ref-kozachok_47-0" class="reference"><a href="#cite_note-kozachok-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p><p>The cube may be regarded as two tetrahedra attached onto the bases of a <a href="/wiki/Triangular_antiprism" class="mw-redirect" title="Triangular antiprism">triangular antiprism</a>.<sup id="cite_ref-FOOTNOTEAlsinaNelsen2015[httpbooksgooglecombooksidFEl2CgAAQBAJpgPA89_89]_48-0" class="reference"><a href="#cite_note-FOOTNOTEAlsinaNelsen2015[httpbooksgooglecombooksidFEl2CgAAQBAJpgPA89_89]-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Representation">Representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=7" title="Edit section: Representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="As_a_graph">As a graph</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=8" title="Edit section: As a graph"><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/Hypercube_graph" title="Hypercube graph">Hypercube graph</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cube_skeleton.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Cube_skeleton.svg/180px-Cube_skeleton.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Cube_skeleton.svg/270px-Cube_skeleton.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Cube_skeleton.svg/360px-Cube_skeleton.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption>The graph of a cube</figcaption></figure> <p>According to <a href="/wiki/Steinitz%27s_theorem" title="Steinitz's theorem">Steinitz's theorem</a>, the <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be represented as the <a href="/wiki/Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">skeleton</a> of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties: <a href="/wiki/Planar_graph" title="Planar graph">planar</a> (the edges of a graph are connected to every vertex without crossing other edges), and <a href="/wiki/K-vertex-connected_graph" title="K-vertex-connected graph">3-connected</a> (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected).<sup id="cite_ref-grunbaum-2003_49-0" class="reference"><a href="#cite_note-grunbaum-2003-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ziegler_50-0" class="reference"><a href="#cite_note-ziegler-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> The skeleton of a cube can be represented as the graph, and it is called the <b>cubical graph</b>, a <a href="/wiki/Platonic_graph" class="mw-redirect" title="Platonic graph">Platonic graph</a>. It has the same number of vertices and edges as the cube, twelve vertices and eight edges.<sup id="cite_ref-rudolph_51-0" class="reference"><a href="#cite_note-rudolph-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> The cubical graph is also classified as a <a href="/wiki/Prism_graph" title="Prism graph">prism graph</a>, resembling the skeleton of a cuboid.<sup id="cite_ref-ps_52-0" class="reference"><a href="#cite_note-ps-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>The cubical graph is a special case of <a href="/wiki/Hypercube_graph" title="Hypercube graph">hypercube graph</a> or <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span>-</span>cube—denoted as <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 Q_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503d0af3998f76cd4eaf8b3cc5e8834e254cb71b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle Q_{n}}" /></span>—because it can be constructed by using the operation known as the <a href="/wiki/Cartesian_product_of_graphs" title="Cartesian product of graphs">Cartesian product of graphs</a>: it involves two graphs connecting the pair of vertices with an edge to form a new graph.<sup id="cite_ref-hh_53-0" class="reference"><a href="#cite_note-hh-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> In the case of the cubical graph, it is the product of two <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 Q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86e8bff64d5e62fc8f45a35875e78bc9bef74a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{2}}" /></span>; roughly speaking, it is a graph resembling a square. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph is <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 Q_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b6c2a91a49263a333768fc2ebebdc379ddf5d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{3}}" /></span>.<sup id="cite_ref-cz_54-0" class="reference"><a href="#cite_note-cz-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> As a part of the hypercube graph, it is also an example of a <a href="/wiki/Unit_distance_graph" title="Unit distance graph">unit distance graph</a>.<sup id="cite_ref-hp_55-0" class="reference"><a href="#cite_note-hp-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>The cubical graph is <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite</a>, meaning every <a href="/wiki/Independent_set_(graph_theory)" title="Independent set (graph theory)">independent set</a> of four vertices can be <a href="/wiki/Disjoint_set" class="mw-redirect" title="Disjoint set">disjoint</a> and the edges connected in those sets.<sup id="cite_ref-berman-graph_56-0" class="reference"><a href="#cite_note-berman-graph-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> However, every vertex in one set cannot connect all vertices in the second, so this bipartite graph is not <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete</a>.<sup id="cite_ref-aw_57-0" class="reference"><a href="#cite_note-aw-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> It is an example of <a href="/wiki/Crown_graph" title="Crown graph">crown graph</a>,<sup id="cite_ref-kl_58-0" class="reference"><a href="#cite_note-kl-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> and of <a href="/wiki/Bipartite_Kneser_graph" class="mw-redirect" title="Bipartite Kneser graph">bipartite Kneser graph</a>.<sup id="cite_ref-berman-graph_56-1" class="reference"><a href="#cite_note-berman-graph-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_orthogonal_projection">In orthogonal projection</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=9" title="Edit section: In orthogonal projection"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called an <a href="/wiki/Orthographic_projection" title="Orthographic projection">orthogonal projection</a>. A polyhedron is considered <i>equiprojective</i> if, for some position of the light, its orthogonal projection is a regular polygon. The cube is equiprojective because, if the light is parallel to one of the four lines joining a vertex to the opposite vertex, its projection is a <a href="/wiki/Regular_hexagon" class="mw-redirect" title="Regular hexagon">regular hexagon</a>.<sup id="cite_ref-hhlnqr_59-0" class="reference"><a href="#cite_note-hhlnqr-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="As_a_configuration_matrix">As a configuration matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=10" title="Edit section: As a configuration matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The cube can be represented as <a href="/wiki/Platonic_solid#As_a_configuration" title="Platonic solid">configuration matrix</a>. A configuration matrix is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The <a href="/wiki/Main_diagonal" title="Main diagonal">diagonal</a> of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. As mentioned above, the cube has eight vertices, twelve edges, and six faces; each element in a matrix's diagonal is denoted as 8, 12, and 6. The first column of the middle row indicates that there are two vertices in (i.e., at the extremes of) each edge, denoted as 2; the middle column of the first row indicates that three edges meet at each vertex, denoted as 3. The following matrix is:<sup id="cite_ref-coxeter_60-0" class="reference"><a href="#cite_note-coxeter-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a76d5e4ed7670e7ea55caa15d1ff5b95542f898" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.899ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Related_figures">Related figures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=11" title="Edit section: Related figures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Construction_of_polyhedra">Construction of polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=12" title="Edit section: Construction of polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:312px;max-width:312px"><div class="trow"><div class="tsingle" style="width:159px;max-width:159px"><div class="thumbimage" style="height:141px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:CubeAndStel.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/CubeAndStel.svg/157px-CubeAndStel.svg.png" decoding="async" width="157" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/CubeAndStel.svg/236px-CubeAndStel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/CubeAndStel.svg/314px-CubeAndStel.svg.png 2x" data-file-width="210" data-file-height="190" /></a></span></div></div><div class="tsingle" style="width:149px;max-width:149px"><div class="thumbimage" style="height:141px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tetrakishexahedron.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/147px-Tetrakishexahedron.jpg" decoding="async" width="147" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/221px-Tetrakishexahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/294px-Tetrakishexahedron.jpg 2x" data-file-width="767" data-file-height="737" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Some of the derived cubes, the <a href="/wiki/Stellated_octahedron" title="Stellated octahedron">stellated octahedron</a> and <a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a>.</div></div></div></div> <p>The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following: </p> <ul><li>When <a href="/wiki/Faceting" title="Faceting">faceting</a> a cube, meaning removing part of the polygonal faces without creating new vertices of a cube, the resulting polyhedron is the <a href="/wiki/Stellated_octahedron" title="Stellated octahedron">stellated octahedron</a>.<sup id="cite_ref-inchbald_61-0" class="reference"><a href="#cite_note-inchbald-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup></li> <li>The cube is <a href="/wiki/Non-composite_polyhedron" class="mw-redirect" title="Non-composite polyhedron">non-composite polyhedron</a>, meaning it is a convex polyhedron that cannot be separated into two or more regular polyhedrons. The cube can be applied to construct a new convex polyhedron by attaching another.<sup id="cite_ref-timofeenko-2010_62-0" class="reference"><a href="#cite_note-timofeenko-2010-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> Attaching a <a href="/wiki/Square_pyramid" title="Square pyramid">square pyramid</a> to each square face of a cube produces its <a href="/wiki/Kleetope" title="Kleetope">Kleetope</a>, a polyhedron known as the <a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a>.<sup id="cite_ref-sod_63-0" class="reference"><a href="#cite_note-sod-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> Suppose one and two equilateral square pyramids are attached to their square faces. In that case, they are the construction of an <a href="/wiki/Elongated_square_pyramid" title="Elongated square pyramid">elongated square pyramid</a> and <a href="/wiki/Elongated_square_bipyramid" title="Elongated square bipyramid">elongated square bipyramid</a> respectively, the <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solid</a>'s examples.<sup id="cite_ref-rajwade_64-0" class="reference"><a href="#cite_note-rajwade-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup></li> <li>Each of the cube's vertices can be <a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">truncated</a>, and the resulting polyhedron is the <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solid</a>, the <a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a>.<sup id="cite_ref-FOOTNOTECromwell1997[httpsbooksgooglecombooksidOJowej1QWpoCpgPA81_81–82]_65-0" class="reference"><a href="#cite_note-FOOTNOTECromwell1997[httpsbooksgooglecombooksidOJowej1QWpoCpgPA81_81–82]-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> When its edges are truncated, it is a <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a>.<sup id="cite_ref-linti_66-0" class="reference"><a href="#cite_note-linti-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> Relatedly, the rhombicuboctahedron can also be constructed by separating the cube's faces and then spreading away, after which adding other triangular and square faces between them; this is known as the "expanded cube". Similarly, it is constructed by the cube's dual, the regular octahedron.<sup id="cite_ref-vxac_67-0" class="reference"><a href="#cite_note-vxac-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup></li> <li>The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in a <a href="/wiki/Trirectangular_tetrahedron" title="Trirectangular tetrahedron">trirectangular tetrahedron</a>.<sup id="cite_ref-FOOTNOTECoxeter1973[httpbooksgooglecombooksid2ee7AQAAQBAJpgPA71_71]_68-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973[httpbooksgooglecombooksid2ee7AQAAQBAJpgPA71_71]-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Snub_cube" title="Snub cube">snub cube</a> is an Archimedean solid that can be constructed by separating away the cube square's face, and filling their gaps with twisted angle equilateral triangles, a process known as <a href="/wiki/Snub_(geometry)" title="Snub (geometry)">snub</a>.<sup id="cite_ref-holme_69-0" class="reference"><a href="#cite_note-holme-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup></li></ul> <p>The cube can be constructed with six <a href="/wiki/Square_pyramid" title="Square pyramid">square pyramids</a>, tiling space by attaching their apices. In some cases, this produces the <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> circumscribing a cube.<sup id="cite_ref-barnes_70-0" class="reference"><a href="#cite_note-barnes-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cundy_71-0" class="reference"><a href="#cite_note-cundy-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Space-filling_and_honeycombs">Space-filling and honeycombs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=13" title="Edit section: Space-filling and honeycombs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hilbert%27s_third_problem" title="Hilbert's third problem">Hilbert's third problem</a> asked whether every two equal volume polyhedra could always be dissected into polyhedral pieces and reassembled into each other. If it was, then the volume of any polyhedron could be defined axiomatically as the volume of an equivalent cube into which it could be reassembled. <a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a> solved this problem in an invention <a href="/wiki/Dehn_invariant" title="Dehn invariant">Dehn invariant</a>, answering that not all polyhedra can be reassembled into a cube.<sup id="cite_ref-gruber_72-0" class="reference"><a href="#cite_note-gruber-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different.<sup id="cite_ref-zeeman_73-0" class="reference"><a href="#cite_note-zeeman-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cubic_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Cubic_honeycomb.png/130px-Cubic_honeycomb.png" decoding="async" width="130" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Cubic_honeycomb.png/195px-Cubic_honeycomb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Cubic_honeycomb.png/260px-Cubic_honeycomb.png 2x" data-file-width="902" data-file-height="900" /></a><figcaption><a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">Cubic honeycomb</a></figcaption></figure> <p>The cube has a Dehn invariant of zero. This indicates the cube is applied for <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">honeycomb</a>. More strongly, the cube is a <a href="/wiki/Space-filling_polyhedron" title="Space-filling polyhedron">space-filling tile</a> in three-dimensional space in which the construction begins by attaching a polyhedron onto its faces without leaving a gap.<sup id="cite_ref-lm_74-0" class="reference"><a href="#cite_note-lm-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> The cube is a <a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedron</a>, a special kind of space-filling polyhedron that can be defined as the <a href="/wiki/Voronoi_cell" class="mw-redirect" title="Voronoi cell">Voronoi cell</a> of a symmetric <a href="/wiki/Delone_set" title="Delone set">Delone set</a>.<sup id="cite_ref-erdahl_75-0" class="reference"><a href="#cite_note-erdahl-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> The plesiohedra include the <a href="/wiki/Parallelohedron" title="Parallelohedron">parallelohedrons</a>, which can be <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translated</a> without rotating to fill a space in which each face of any of its copies is attached to a like face of another copy. There are five kinds of parallelohedra, one of which is the cuboid.<sup id="cite_ref-alexandrov_76-0" class="reference"><a href="#cite_note-alexandrov-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> Every three-dimensional parallelohedron is <a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a>, a <a href="/wiki/Centrally_symmetric" class="mw-redirect" title="Centrally symmetric">centrally symmetric</a> polyhedron whose faces are <a href="/wiki/Zonogon" title="Zonogon">centrally symmetric polygons</a>.<sup id="cite_ref-shephard_77-0" class="reference"><a href="#cite_note-shephard-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> In the case of cube, it can be represented as the <a href="/wiki/Cell_(geometry)" class="mw-redirect" title="Cell (geometry)">cell</a>. Some honeycombs have cubes as the only cells. One example is <a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">cubic honeycomb</a>, the only proper honeycomb with four cubes around every edge.<sup id="cite_ref-twelveessay_78-0" class="reference"><a href="#cite_note-twelveessay-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ns_79-0" class="reference"><a href="#cite_note-ns-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Net_of_tesseract.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Net_of_tesseract.gif/130px-Net_of_tesseract.gif" decoding="async" width="130" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Net_of_tesseract.gif/195px-Net_of_tesseract.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Net_of_tesseract.gif/260px-Net_of_tesseract.gif 2x" data-file-width="355" data-file-height="400" /></a><figcaption><a href="/wiki/Dali_cross" class="mw-redirect" title="Dali cross">Dali cross</a>, the net of a <a href="/wiki/Tesseract" title="Tesseract">tesseract</a></figcaption></figure> <p><a href="/wiki/Polycube" title="Polycube">Polycube</a> is a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as the <a href="/wiki/Polyominoes" class="mw-redirect" title="Polyominoes">polyominoes</a> in three-dimensional space.<sup id="cite_ref-lunnon_80-0" class="reference"><a href="#cite_note-lunnon-80"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> When four cubes are stacked vertically, and the other four are attached to the second-from-top cube of the stack, the resulting polycube is <a href="/wiki/Dali_cross" class="mw-redirect" title="Dali cross">Dali cross</a>, after <a href="/wiki/Salvador_Dali" class="mw-redirect" title="Salvador Dali">Salvador Dali</a>. In addition to popular cultures, the Dali cross is a tile space polyhedron,<sup id="cite_ref-hut_81-0" class="reference"><a href="#cite_note-hut-81"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pucc_82-0" class="reference"><a href="#cite_note-pucc-82"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> which can be represented as the net of a <a href="/wiki/Tesseract" title="Tesseract">tesseract</a>. A tesseract is a cube analogous' <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">four-dimensional space</a> bounded by twenty-four squares and eight cubes.<sup id="cite_ref-hall_83-0" class="reference"><a href="#cite_note-hall-83"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Miscellaneous">Miscellaneous</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cube&action=edit&section=14" title="Edit section: Miscellaneous"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti" /><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:351px;max-width:351px"><div class="trow"><div class="tsingle" style="width:115px;max-width:115px"><div class="thumbimage" style="height:113px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:UC07-6_cubes.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/UC07-6_cubes.png/113px-UC07-6_cubes.png" decoding="async" width="113" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/UC07-6_cubes.png/170px-UC07-6_cubes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/UC07-6_cubes.png/226px-UC07-6_cubes.png 2x" data-file-width="800" data-file-height="800" /></a></span></div></div><div class="tsingle" style="width:115px;max-width:115px"><div class="thumbimage" style="height:113px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:UC08-3_cubes.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/UC08-3_cubes.png/113px-UC08-3_cubes.png" decoding="async" width="113" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/UC08-3_cubes.png/170px-UC08-3_cubes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/UC08-3_cubes.png/226px-UC08-3_cubes.png 2x" data-file-width="800" data-file-height="800" /></a></span></div></div><div class="tsingle" style="width:115px;max-width:115px"><div class="thumbimage" style="height:113px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:UC09-5_cubes.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/UC09-5_cubes.png/113px-UC09-5_cubes.png" decoding="async" width="113" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/UC09-5_cubes.png/170px-UC09-5_cubes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/UC09-5_cubes.png/226px-UC09-5_cubes.png 2x" data-file-width="800" data-file-height="800" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Enumeration according to <a href="#CITEREFSkilling1976">Skilling (1976)</a>: compound of six cubes with rotational freedom <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 \mathrm {UC} _{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {UC} _{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e0f33e098a071143ab4fab208e0c743e4636c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle \mathrm {UC} _{7}}" /></span>, <a href="/wiki/Compound_of_three_cubes" title="Compound of three cubes">three cubes</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {UC} _{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {UC} _{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3894305bec0824b6d95dadf244c1fe5169cf8ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle \mathrm {UC} _{8}}" /></span>, and five cubes <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 \mathrm {UC} _{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {UC} _{9}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc1ba01a24101b815f660c2e0dae7e0b63bcc70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.475ex; height:2.509ex;" alt="{\displaystyle \mathrm {UC} _{9}}" /></span></div></div></div></div> <p><span class="anchor" id="Compound_of_cubes"></span>Compound of cubes is the <a href="/wiki/Polyhedral_compound" class="mw-redirect" title="Polyhedral compound">polyhedral compounds</a> in which the cubes are sharing the same centre. They belong to the <a href="/wiki/Uniform_polyhedron_compound" title="Uniform polyhedron compound">uniform polyhedron compound</a>, meaning they are polyhedral compounds whose constituents are identical (although possibly <a href="/wiki/Enantiomorphous" class="mw-redirect" title="Enantiomorphous">enantiomorphous</a>) <a href="/wiki/Uniform_polyhedra" class="mw-redirect" title="Uniform polyhedra">uniform polyhedra</a>, in an arrangement that is also uniform. The list of compounds enumerated by <a href="#CITEREFSkilling1976">Skilling (1976)</a> in seventh to ninth uniform compound for the compound of six cubes with rotational freedom, <a href="/wiki/Compound_of_three_cubes" title="Compound of three cubes">three cubes</a>, and five cubes respectively.<sup id="cite_ref-skilling_84-0" class="reference"><a href="#cite_note-skilling-84"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> Two compounds, consisting of <a href="/wiki/Compound_of_two_cubes" class="mw-redirect" title="Compound of two cubes">two</a> and three cubes were found in <a href="/wiki/M._C._Escher" title="M. C. Escher">Escher</a>'s <a href="/wiki/Wood_engraving" title="Wood engraving">wood engraving</a> print <a href="/wiki/Stars_(M._C._Escher)" title="Stars (M. C. Escher)"><i>Stars</i></a> and <a href="/wiki/Max_Br%C3%BCckner" title="Max Brückner">Max Brückner</a>'s book <i>Vielecke und Vielflache</i>.<sup id="cite_ref-hart_85-0" class="reference"><a href="#cite_note-hart-85"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Square_on_sphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/130px-Square_on_sphere.svg.png" decoding="async" width="130" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/195px-Square_on_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Square_on_sphere.svg/260px-Square_on_sphere.svg.png 2x" data-file-width="816" data-file-height="815" /></a><figcaption>Spherical cube</figcaption></figure> <p><span class="anchor" id="Spherical_cube"></span>The spherical cube represents the <a href="/wiki/Spherical_polyhedron" title="Spherical polyhedron">spherical polyhedron</a>, consisting of six spherical squares with 120° interior angle on each vertex. It has <a href="/wiki/Vector_equilibrium" class="mw-redirect" title="Vector equilibrium">vector equilibrium</a>, meaning that the distance from the centroid and each vertex is the same as the distance from that and each edge.<sup id="cite_ref-popko_86-0" class="reference"><a href="#cite_note-popko-86"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-fuller_87-0" class="reference"><a href="#cite_note-fuller-87"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> Its dual is the <a href="/wiki/Spherical_octahedron" class="mw-redirect" title="Spherical octahedron">spherical octahedron</a>. The spherical cube can be modeled by the <a href="/wiki/Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arc</a> of <a href="/wiki/Great_circle" title="Great circle">great circles</a>, creating bounds as the edges of a <a href="/wiki/Spherical_polygon" class="mw-redirect" title="Spherical polygon">spherical square</a>.<sup id="cite_ref-yackel_88-0" class="reference"><a href="#cite_note-yackel-88"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> </p><p>The topological object <a href="/wiki/3-torus" title="3-torus">three-dimensional torus</a> is a topological space defined to be <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the Cartesian product of three circles. It can be represented as a three-dimensional model of the cube shape.<sup id="cite_ref-marar_89-0" class="reference"><a href="#cite_note-marar-89"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> </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=Cube&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bhargava_cube" title="Bhargava cube">Bhargava cube</a>, a configuration to study the law of <a href="/wiki/Binary_quadratic_form" title="Binary quadratic form">binary quadratic form</a> and other such forms, of which the cube's vertices represent the <a href="/wiki/Integer" title="Integer">integer</a>.</li> <li><a href="/wiki/Hemicube_(geometry)" title="Hemicube (geometry)">Hemicube</a>, a polyhedron produced by cutting a cube in half with a plane.</li> <li><a href="/wiki/Squaring_the_square" title="Squaring the square">Squaring the square</a>'s three-dimensional analogue, <a href="/wiki/Cubing_the_cube" class="mw-redirect" title="Cubing the cube">cubing the cube</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=Cube&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-mk-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-mk_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMillsKolf1999" class="citation book cs1">Mills, Steve; Kolf, Hillary (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dvFfTAR6XwEC&pg=PA16"><i>Maths Dictionary</i></a>. Heinemann. p. 16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-435-02474-1" title="Special:BookSources/978-0-435-02474-1"><bdi>978-0-435-02474-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maths+Dictionary&rft.pages=16&rft.pub=Heinemann&rft.date=1999&rft.isbn=978-0-435-02474-1&rft.aulast=Mills&rft.aufirst=Steve&rft.au=Kolf%2C+Hillary&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdvFfTAR6XwEC%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-johnson-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-johnson_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJohnson1966" class="citation journal cs1"><a href="/wiki/Norman_W._Johnson" class="mw-redirect" title="Norman W. Johnson">Johnson, Norman W.</a> (1966). <a rel="nofollow" class="external text" href="https://doi.org/10.4153%2Fcjm-1966-021-8">"Convex polyhedra with regular faces"</a>. <i><a href="/wiki/Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a></i>. <b>18</b>: <span class="nowrap">169–</span>200. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4153%2Fcjm-1966-021-8">10.4153/cjm-1966-021-8</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0185507">0185507</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122006114">122006114</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0132.14603">0132.14603</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=Convex+polyhedra+with+regular+faces&rft.volume=18&rft.pages=%3Cspan+class%3D%22nowrap%22%3E169-%3C%2Fspan%3E200&rft.date=1966&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0132.14603%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0185507%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122006114%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4153%2Fcjm-1966-021-8&rft.aulast=Johnson&rft.aufirst=Norman+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.4153%252Fcjm-1966-021-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span> See table II, line 3.</span> </li> <li id="cite_note-hs-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-hs_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHerrmannSally2013" class="citation book cs1">Herrmann, Diane L.; Sally, Paul J. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b2fjR81h6yEC&pg=PA252"><i>Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory</i></a>. Taylor & Francis. p. 252. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4665-5464-1" title="Special:BookSources/978-1-4665-5464-1"><bdi>978-1-4665-5464-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number%2C+Shape%2C+%26+Symmetry%3A+An+Introduction+to+Number+Theory%2C+Geometry%2C+and+Group+Theory&rft.pages=252&rft.pub=Taylor+%26+Francis&rft.date=2013&rft.isbn=978-1-4665-5464-1&rft.aulast=Herrmann&rft.aufirst=Diane+L.&rft.au=Sally%2C+Paul+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Db2fjR81h6yEC%26pg%3DPA252&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-jessen-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-jessen_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJessen1967" class="citation journal cs1"><a href="/wiki/B%C3%B8rge_Jessen" title="Børge Jessen">Jessen, Børge</a> (1967). "Orthogonal icosahedra". <i>Nordisk Matematisk Tidskrift</i>. <b>15</b> (2): <span class="nowrap">90–</span>96. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24524998">24524998</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0226494">0226494</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nordisk+Matematisk+Tidskrift&rft.atitle=Orthogonal+icosahedra&rft.volume=15&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E90-%3C%2Fspan%3E96&rft.date=1967&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24524998%23id-name%3DJSTOR&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0226494%23id-name%3DMR&rft.aulast=Jessen&rft.aufirst=B%C3%B8rge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-calter-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-calter_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCalterCalter2011" class="citation book cs1">Calter, Paul; Calter, Michael (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4fHwTZK3JEIC&pg=PA197"><i>Technical Mathematics</i></a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. p. 197. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-53492-2" title="Special:BookSources/978-0-470-53492-2"><bdi>978-0-470-53492-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Technical+Mathematics&rft.pages=197&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.isbn=978-0-470-53492-2&rft.aulast=Calter&rft.aufirst=Paul&rft.au=Calter%2C+Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4fHwTZK3JEIC%26pg%3DPA197&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-khattar-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-khattar_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-khattar_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhattar2008" class="citation book cs1">Khattar, Dinesh (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BoBCOYuOlzkC&pg=PA377"><i>Guide to Objective Arithmetic</i></a> (2nd ed.). <a href="/wiki/Pearson_Education" title="Pearson Education">Pearson Education</a>. p. 377. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-317-1682-3" title="Special:BookSources/978-81-317-1682-3"><bdi>978-81-317-1682-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guide+to+Objective+Arithmetic&rft.pages=377&rft.edition=2nd&rft.pub=Pearson+Education&rft.date=2008&rft.isbn=978-81-317-1682-3&rft.aulast=Khattar&rft.aufirst=Dinesh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBoBCOYuOlzkC%26pg%3DPA377&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-thomson-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-thomson_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThomson1845" class="citation book cs1"><a href="/wiki/James_Thomson_(mathematician)" title="James Thomson (mathematician)">Thomson, James</a> (1845). <a rel="nofollow" class="external text" href="https://archive.org/details/anelementarytre01thomgoog/page/n15"><i>An Elementary Treatise on Algebra: Theoretical and Practical</i></a>. London: Longman, Brown, Green, and Longmans. p. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Elementary+Treatise+on+Algebra%3A+Theoretical+and+Practical&rft.place=London&rft.pages=4&rft.pub=Longman%2C+Brown%2C+Green%2C+and+Longmans&rft.date=1845&rft.aulast=Thomson&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fanelementarytre01thomgoog%2Fpage%2Fn15&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-ball-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-ball_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBall2010" class="citation book cs1">Ball, Keith (2010). "High-dimensional geometry and its probabilistic analogues". In <a href="/wiki/Timothy_Gowers" title="Timothy Gowers">Gowers, Timothy</a> (ed.). <a href="/wiki/The_Princeton_Companion_to_Mathematics" title="The Princeton Companion to Mathematics"><i>The Princeton Companion to Mathematics</i></a>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA671">671</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781400830398" title="Special:BookSources/9781400830398"><bdi>9781400830398</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=High-dimensional+geometry+and+its+probabilistic+analogues&rft.btitle=The+Princeton+Companion+to+Mathematics&rft.pages=671&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=9781400830398&rft.aulast=Ball&rft.aufirst=Keith&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-hr-w-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-hr-w_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1"><i>Geometry: Reteaching Masters</i>. Holt Rinehart & Winston. 2001. p. 74. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780030543289" title="Special:BookSources/9780030543289"><bdi>9780030543289</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+Reteaching+Masters&rft.pages=74&rft.pub=Holt+Rinehart+%26+Winston&rft.date=2001&rft.isbn=9780030543289&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-sriraman-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-sriraman_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSriraman2009" class="citation book cs1">Sriraman, Bharath (2009). "Mathematics and literature (the sequel): imagination as a pathway to advanced mathematical ideas and philosophy". In Sriraman, Bharath; Freiman, Viktor; Lirette-Pitre, Nicole (eds.). <i>Interdisciplinarity, Creativity, and Learning: Mathematics With Literature, Paradoxes, History, Technology, and Modeling</i>. The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education. Vol. 7. Information Age Publishing, Inc. pp. <span class="nowrap">41–</span>54. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781607521013" title="Special:BookSources/9781607521013"><bdi>9781607521013</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+and+literature+%28the+sequel%29%3A+imagination+as+a+pathway+to+advanced+mathematical+ideas+and+philosophy&rft.btitle=Interdisciplinarity%2C+Creativity%2C+and+Learning%3A+Mathematics+With+Literature%2C+Paradoxes%2C+History%2C+Technology%2C+and+Modeling&rft.series=The+Montana+Mathematics+Enthusiast%3A+Monograph+Series+in+Mathematics+Education&rft.pages=%3Cspan+class%3D%22nowrap%22%3E41-%3C%2Fspan%3E54&rft.pub=Information+Age+Publishing%2C+Inc.&rft.date=2009&rft.isbn=9781607521013&rft.aulast=Sriraman&rft.aufirst=Bharath&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-jwy-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-jwy_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJerrardWetzelYuan2017" class="citation journal cs1">Jerrard, Richard P.; Wetzel, John E.; Yuan, Liping (April 2017). "Platonic passages". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>90</b> (2). Washington, DC: <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>: <span class="nowrap">87–</span>98. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Fmath.mag.90.2.87">10.4169/math.mag.90.2.87</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:218542147">218542147</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Platonic+passages&rft.volume=90&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E87-%3C%2Fspan%3E98&rft.date=2017-04&rft_id=info%3Adoi%2F10.4169%2Fmath.mag.90.2.87&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A218542147%23id-name%3DS2CID&rft.aulast=Jerrard&rft.aufirst=Richard+P.&rft.au=Wetzel%2C+John+E.&rft.au=Yuan%2C+Liping&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-lutzen-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-lutzen_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLützen2010" class="citation journal cs1">Lützen, Jesper (2010). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.2009.00160.x">"The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle"</a>. <i>Centaurus</i>. <b>52</b> (1): <span class="nowrap">4–</span>37. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1600-0498.2009.00160.x">10.1111/j.1600-0498.2009.00160.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Centaurus&rft.atitle=The+Algebra+of+Geometric+Impossibility%3A+Descartes+and+Montucla+on+the+Impossibility+of+the+Duplication+of+the+Cube+and+the+Trisection+of+the+Angle&rft.volume=52&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4-%3C%2Fspan%3E37&rft.date=2010&rft_id=info%3Adoi%2F10.1111%2Fj.1600-0498.2009.00160.x&rft.aulast=L%C3%BCtzen&rft.aufirst=Jesper&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0498.2009.00160.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-radii-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-radii_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1973">Coxeter (1973)</a> Table I(i), pp. 292–293. See the columns labeled <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 {}_{0}\!\mathrm {R} /\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{0}\!\mathrm {R} /\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0147877a0351e449db06f72b4d475e001f900e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.51ex; height:2.843ex;" alt="{\displaystyle {}_{0}\!\mathrm {R} /\ell }" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{1}\!\mathrm {R} /\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{1}\!\mathrm {R} /\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064c055a3b71309163d3bd09f2c9c40a95a4d15a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.51ex; height:2.843ex;" alt="{\displaystyle {}_{1}\!\mathrm {R} /\ell }" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}\!\mathrm {R} /\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{2}\!\mathrm {R} /\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73edbf197f36ba2d80fe9cc902b6ab537b75f129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.51ex; height:2.843ex;" alt="{\displaystyle {}_{2}\!\mathrm {R} /\ell }" /></span>, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a443fec94d4ffc53fd40d50d2c882927d08b0b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle 2\ell }" /></span> as the edge length (see p. 2).</span> </li> <li id="cite_note-poo-sung-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-poo-sung_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPoo-Sung2016" class="citation journal cs1">Poo-Sung, Park, Poo-Sung (2016). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161010184811/http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf">"Regular polytope distances"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Forum_Geometricorum" title="Forum Geometricorum">Forum Geometricorum</a></i>. <b>16</b>: <span class="nowrap">227–</span>232. Archived from <a rel="nofollow" class="external text" href="http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-10-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-05-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Forum+Geometricorum&rft.atitle=Regular+polytope+distances&rft.volume=16&rft.pages=%3Cspan+class%3D%22nowrap%22%3E227-%3C%2Fspan%3E232&rft.date=2016&rft.aulast=Poo-Sung&rft.aufirst=Park%2C+Poo-Sung&rft_id=http%3A%2F%2Fforumgeom.fau.edu%2FFG2016volume16%2FFG201627.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-french-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-french_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFrench1988" class="citation journal cs1">French, Doug (1988). "Reflections on a Cube". <i>Mathematics in School</i>. <b>17</b> (4): <span class="nowrap">30–</span>33. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30214515">30214515</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+in+School&rft.atitle=Reflections+on+a+Cube&rft.volume=17&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E30-%3C%2Fspan%3E33&rft.date=1988&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F30214515%23id-name%3DJSTOR&rft.aulast=French&rft.aufirst=Doug&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-cromwell-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-cromwell_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCromwell1997" class="citation book cs1">Cromwell, Peter R. (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom/page/309"><i>Polyhedra</i></a>. Cambridge University Press. p. 309. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55432-9" title="Special:BookSources/978-0-521-55432-9"><bdi>978-0-521-55432-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyhedra&rft.pages=309&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-55432-9&rft.aulast=Cromwell&rft.aufirst=Peter+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpolyhedra0000crom%2Fpage%2F309&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-cp-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-cp_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCunninghamPellicer2024" class="citation journal cs1">Cunningham, Gabe; Pellicer, Daniel (2024). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs40590-024-00600-z">"Finite 3-orbit polyhedra in ordinary space, II"</a>. <i>Boletín de la Sociedad Matemática Mexicana</i>. <b>30</b> (32). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs40590-024-00600-z">10.1007/s40590-024-00600-z</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bolet%C3%ADn+de+la+Sociedad+Matem%C3%A1tica+Mexicana&rft.atitle=Finite+3-orbit+polyhedra+in+ordinary+space%2C+II&rft.volume=30&rft.issue=32&rft.date=2024&rft_id=info%3Adoi%2F10.1007%2Fs40590-024-00600-z&rft.aulast=Cunningham&rft.aufirst=Gabe&rft.au=Pellicer%2C+Daniel&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs40590-024-00600-z&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span> See p. 276.</span> </li> <li id="cite_note-cr-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-cr_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCundyRollett1961" class="citation book cs1"><a href="/wiki/Martyn_Cundy" title="Martyn Cundy">Cundy, H. Martyn</a>; Rollett, A.P. (1961). "3.2 Duality". <a href="/wiki/Mathematical_Models_(Cundy_and_Rollett)" title="Mathematical Models (Cundy and Rollett)"><i>Mathematical models</i></a> (2nd ed.). Oxford: Clarendon Press. pp. <span class="nowrap">78–</span>79. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0124167">0124167</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.2+Duality&rft.btitle=Mathematical+models&rft.place=Oxford&rft.pages=%3Cspan+class%3D%22nowrap%22%3E78-%3C%2Fspan%3E79&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1961&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0124167%23id-name%3DMR&rft.aulast=Cundy&rft.aufirst=H.+Martyn&rft.au=Rollett%2C+A.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-erickson-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-erickson_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFErickson2011" class="citation book cs1">Erickson, Martin (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62"><i>Beautiful Mathematics</i></a>. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. p. 62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61444-509-8" title="Special:BookSources/978-1-61444-509-8"><bdi>978-1-61444-509-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beautiful+Mathematics&rft.pages=62&rft.pub=Mathematical+Association+of+America&rft.date=2011&rft.isbn=978-1-61444-509-8&rft.aulast=Erickson&rft.aufirst=Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLgeP62-ZxikC%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-mclean-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-mclean_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mclean_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMcLean1990" class="citation journal cs1">McLean, K. Robin (1990). "Dungeons, dragons, and dice". <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>. <b>74</b> (469): <span class="nowrap">243–</span>256. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3619822">10.2307/3619822</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3619822">3619822</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:195047512">195047512</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Dungeons%2C+dragons%2C+and+dice&rft.volume=74&rft.issue=469&rft.pages=%3Cspan+class%3D%22nowrap%22%3E243-%3C%2Fspan%3E256&rft.date=1990&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A195047512%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3619822%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3619822&rft.aulast=McLean&rft.aufirst=K.+Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span> See p. 247.</span> </li> <li id="cite_note-grunbaum-1997-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-grunbaum-1997_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrünbaum1997" class="citation journal cs1"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a> (1997). "Isogonal Prismatoids". <i>Discrete & Computational Geometry</i>. <b>18</b> (1): <span class="nowrap">13–</span>52. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FPL00009307">10.1007/PL00009307</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+%26+Computational+Geometry&rft.atitle=Isogonal+Prismatoids&rft.volume=18&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E13-%3C%2Fspan%3E52&rft.date=1997&rft_id=info%3Adoi%2F10.1007%2FPL00009307&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-senechal-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-senechal_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSenechal1989" class="citation book cs1">Senechal, Marjorie (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12">"A Brief Introduction to Tilings"</a>. In Jarić, Marko (ed.). <i>Introduction to the Mathematics of Quasicrystals</i>. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. p. 12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+Brief+Introduction+to+Tilings&rft.btitle=Introduction+to+the+Mathematics+of+Quasicrystals&rft.pages=12&rft.pub=Academic+Press&rft.date=1989&rft.aulast=Senechal&rft.aufirst=Marjorie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOToVjZW9CKMC%26pg%3DPA12&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-wd-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-wd_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWalterDeloudi2009" class="citation book cs1">Walter, Steurer; Deloudi, Sofia (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nVx-tu596twC&pg=PA50"><i>Crystallography of Quasicrystals: Concepts, Methods and Structures</i></a>. Springer Series in Materials Science. 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The Johns Hopkins University Press. p. 76. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-9012-3" title="Special:BookSources/978-0-8018-9012-3"><bdi>978-0-8018-9012-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Adventures+in+Group+Theory%3A+Rubik%27s+Cube%2C+Merlin%27s+Machine%2C+and+Other+Mathematical+Toys&rft.pages=76&rft.edition=2nd&rft.pub=The+Johns+Hopkins+University+Press&rft.date=2008&rft.isbn=978-0-8018-9012-3&rft.aulast=Joyner&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6an_DwAAQBAJ%26pg%3DPA76&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-moore-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-moore_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMoore2018" class="citation journal cs1">Moore, Kimberly (2018). "Minecraft Comes to Math Class". <i>Mathematics Teaching in the Middle School</i>. <b>23</b> (6): <span class="nowrap">334–</span>341. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fmathteacmiddscho.23.6.0334">10.5951/mathteacmiddscho.23.6.0334</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/10.5951/mathteacmiddscho.23.6.0334">10.5951/mathteacmiddscho.23.6.0334</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Teaching+in+the+Middle+School&rft.atitle=Minecraft+Comes+to+Math+Class&rft.volume=23&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E334-%3C%2Fspan%3E341&rft.date=2018&rft_id=info%3Adoi%2F10.5951%2Fmathteacmiddscho.23.6.0334&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F10.5951%2Fmathteacmiddscho.23.6.0334%23id-name%3DJSTOR&rft.aulast=Moore&rft.aufirst=Kimberly&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-rz-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-rz_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFReavenZeilten2006" class="citation book cs1">Reaven, Marci; Zeilten, Steve (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xgOtV4Nynd0C&pg=PA77"><i>Hidden New York: A Guide to Places that Matter</i></a>. Rutgers University Press. p. 77. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8135-3890-7" title="Special:BookSources/978-0-8135-3890-7"><bdi>978-0-8135-3890-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hidden+New+York%3A+A+Guide+to+Places+that+Matter&rft.pages=77&rft.pub=Rutgers+University+Press&rft.date=2006&rft.isbn=978-0-8135-3890-7&rft.aulast=Reaven&rft.aufirst=Marci&rft.au=Zeilten%2C+Steve&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxgOtV4Nynd0C%26pg%3DPA77&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-barrow-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-barrow_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJohn_D._Barrow1999" class="citation book cs1"><a href="/wiki/John_D._Barrow" title="John D. 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"Renaissance mathematics and architectural proportion in Alberti's De re aedificatoria". <i>Architectural Research Quarterly</i>. <b>2</b> (1): <span class="nowrap">54–</span>65. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS135913550000110X">10.1017/S135913550000110X</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:110346888">110346888</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Architectural+Research+Quarterly&rft.atitle=Renaissance+mathematics+and+architectural+proportion+in+Alberti%27s+De+re+aedificatoria&rft.volume=2&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E54-%3C%2Fspan%3E65&rft.date=1996&rft_id=info%3Adoi%2F10.1017%2FS135913550000110X&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A110346888%23id-name%3DS2CID&rft.aulast=March&rft.aufirst=Lionel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-an-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-an_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAlsinaNelsen2015" class="citation book cs1">Alsina, Claudi; Nelsen, Roger B. 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Vol. 50. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. p. 85. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61444-216-5" title="Special:BookSources/978-1-61444-216-5"><bdi>978-1-61444-216-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+Space+Odyssey%3A+Solid+Geometry+in+the+21st+Century&rft.pages=85&rft.pub=Mathematical+Association+of+America&rft.date=2015&rft.isbn=978-1-61444-216-5&rft.aulast=Alsina&rft.aufirst=Claudi&rft.au=Nelsen%2C+Roger+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFEl2CgAAQBAJ%26pg%3DPA85&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-tisza-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-tisza_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTisza2001" class="citation book cs1">Tisza, Miklós (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=y1eTDQRdI2wC&pg=PA45"><i>Physical Metallurgy for Engineers</i></a>. <a href="/wiki/Materials_Park,_Ohio" class="mw-redirect" title="Materials Park, Ohio">Materials Park, Ohio</a>: <a href="/wiki/ASM_International_(society)" title="ASM International (society)">ASM International</a>. p. 45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61503-241-9" title="Special:BookSources/978-1-61503-241-9"><bdi>978-1-61503-241-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Metallurgy+for+Engineers&rft.place=Materials+Park%2C+Ohio&rft.pages=45&rft.pub=ASM+International&rft.date=2001&rft.isbn=978-1-61503-241-9&rft.aulast=Tisza&rft.aufirst=Mikl%C3%B3s&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dy1eTDQRdI2wC%26pg%3DPA45&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-hoffmann-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-hoffmann_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHoffmann2020" class="citation book cs1">Hoffmann, Frank (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=16H0DwAAQBAJ&pg=PA35"><i>Introduction to Crystallography</i></a>. Springer. p. 35. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-35110-6">10.1007/978-3-030-35110-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-35110-6" title="Special:BookSources/978-3-030-35110-6"><bdi>978-3-030-35110-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Crystallography&rft.pages=35&rft.pub=Springer&rft.date=2020&rft_id=info%3Adoi%2F10.1007%2F978-3-030-35110-6&rft.isbn=978-3-030-35110-6&rft.aulast=Hoffmann&rft.aufirst=Frank&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D16H0DwAAQBAJ%26pg%3DPA35&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-biegasiewicz-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-biegasiewicz_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBiegasiewiczGriffithsSavageTsanakstidis2015" class="citation journal cs1">Biegasiewicz, Kyle; Griffiths, Justin; Savage, G. Paul; Tsanakstidis, John; Priefer, Ronny (2015). "Cubane: 50 years later". <i>Chemical Reviews</i>. <b>115</b> (14): <span class="nowrap">6719–</span>6745. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fcr500523x">10.1021/cr500523x</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26102302">26102302</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Reviews&rft.atitle=Cubane%3A+50+years+later&rft.volume=115&rft.issue=14&rft.pages=%3Cspan+class%3D%22nowrap%22%3E6719-%3C%2Fspan%3E6745&rft.date=2015&rft_id=info%3Adoi%2F10.1021%2Fcr500523x&rft_id=info%3Apmid%2F26102302&rft.aulast=Biegasiewicz&rft.aufirst=Kyle&rft.au=Griffiths%2C+Justin&rft.au=Savage%2C+G.+Paul&rft.au=Tsanakstidis%2C+John&rft.au=Priefer%2C+Ronny&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-haeckel-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-haeckel_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHaeckel1904" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Ernst_Haeckel" title="Ernst Haeckel">Haeckel, E.</a> (1904). <i><a href="/wiki/Kunstformen_der_Natur" title="Kunstformen der Natur">Kunstformen der Natur</a></i> (in German).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kunstformen+der+Natur&rft.date=1904&rft.aulast=Haeckel&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span> See <a rel="nofollow" class="external text" href="http://www.biolib.de/haeckel/kunstformen/index.html">here</a> for an online book.</span> </li> <li id="cite_note-knstv-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-knstv_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKov´acsNagyStomfaiTurgay2021" class="citation journal cs1">Kov´acs, Gergely; Nagy, Benedek Nagy; Stomfai, Gergely; Turgay, Nes¸et Deni̇z; Vizv´ari, B´ela (2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2F2021%2F5582034">"On Chamfer Distances on the Square and Body-Centered CubicGrids: An Operational Research Approach"</a>. <i>Mathematical Problems in Engineering</i>: <span class="nowrap">1–</span>9. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1155%2F2021%2F5582034">10.1155/2021/5582034</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Problems+in+Engineering&rft.atitle=On+Chamfer+Distances+on+the+Square+and+Body-Centered+CubicGrids%3A+An+Operational+Research+Approach&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E9&rft.date=2021&rft_id=info%3Adoi%2F10.1155%2F2021%2F5582034&rft.aulast=Kov%C2%B4acs&rft.aufirst=Gergely&rft.au=Nagy%2C+Benedek+Nagy&rft.au=Stomfai%2C+Gergely&rft.au=Turgay%2C+Nes%C2%B8et+Deni%CC%87z&rft.au=Vizv%C2%B4ari%2C+B%C2%B4ela&rft_id=https%3A%2F%2Fdoi.org%2F10.1155%252F2021%252F5582034&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-cmsi-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-cmsi_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChinMohamedShariffIshikawa2021" class="citation conference cs1">Chin, Daniel Jie Yuan Chin; Mohamed, Ahmad Sufril Azlan; Shariff, Khairul Anuar; Ishikawa, Kunio (23–25 November 2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GclOEAAAQBAJ&pg=PA376">"GPU-Accelerated Enhanced Marching Cubes 33 for Fast 3D Reconstruction of Large Bone Defect CT Images"</a>. 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Newnes. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-096529-1" title="Special:BookSources/978-0-08-096529-1"><bdi>978-0-08-096529-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Catenated+Compounds+-+Group+13+%5BAl%2C+Ga%2C+In%2C+Tl%5D&rft.btitle=Comprehensive+Inorganic+Chemistry+II%3A+From+Elements+to+Applications&rft.pages=41&rft.pub=Newnes&rft.date=2013&rft.isbn=978-0-08-096529-1&rft.aulast=Linti&rft.aufirst=G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_4C7oid1kQQC%26pg%3DRA7-PA41&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-vxac-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-vxac_67-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVianaXavierAiresCampos2019" class="citation book cs1">Viana, Vera; Xavier, João Pedro; Aires, Ana Paula; Campos, Helena (2019). "Interactive Expansion of Achiral Polyhedra". In Cocchiarella, Luigi (ed.). <i>ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary - Milan, Italy, August 3-7, 2018</i>. Advances in Intelligent Systems and Computing. Vol. 809. Springer. p. 1123. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-95588-9">10.1007/978-3-319-95588-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-95587-2" title="Special:BookSources/978-3-319-95587-2"><bdi>978-3-319-95587-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Interactive+Expansion+of+Achiral+Polyhedra&rft.btitle=ICGG+2018+-+Proceedings+of+the+18th+International+Conference+on+Geometry+and+Graphics+40th+Anniversary+-+Milan%2C+Italy%2C+August+3-7%2C+2018&rft.series=Advances+in+Intelligent+Systems+and+Computing&rft.pages=1123&rft.pub=Springer&rft.date=2019&rft_id=info%3Adoi%2F10.1007%2F978-3-319-95588-9&rft.isbn=978-3-319-95587-2&rft.aulast=Viana&rft.aufirst=Vera&rft.au=Xavier%2C+Jo%C3%A3o+Pedro&rft.au=Aires%2C+Ana+Paula&rft.au=Campos%2C+Helena&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span> See Fig. 6.</span> </li> <li id="cite_note-FOOTNOTECoxeter1973[httpbooksgooglecombooksid2ee7AQAAQBAJpgPA71_71]-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeter1973[httpbooksgooglecombooksid2ee7AQAAQBAJpgPA71_71]_68-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeter1973">Coxeter (1973)</a>, p. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=2ee7AQAAQBAJ&pg=PA71">71</a>.</span> </li> <li id="cite_note-holme-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-holme_69-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHolme2010" class="citation book cs1">Holme, A. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zXwQGo8jyHUC"><i>Geometry: Our Cultural Heritage</i></a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-14441-7">10.1007/978-3-642-14441-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-14441-7" title="Special:BookSources/978-3-642-14441-7"><bdi>978-3-642-14441-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+Our+Cultural+Heritage&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-3-642-14441-7&rft.isbn=978-3-642-14441-7&rft.aulast=Holme&rft.aufirst=A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzXwQGo8jyHUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-barnes-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-barnes_70-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarnes2012" class="citation book cs1">Barnes, John (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA82"><i>Gems of Geometry</i></a> (2nd ed.). Springer. p. 82. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-30964-9">10.1007/978-3-642-30964-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-30964-9" title="Special:BookSources/978-3-642-30964-9"><bdi>978-3-642-30964-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gems+of+Geometry&rft.pages=82&rft.edition=2nd&rft.pub=Springer&rft.date=2012&rft_id=info%3Adoi%2F10.1007%2F978-3-642-30964-9&rft.isbn=978-3-642-30964-9&rft.aulast=Barnes&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7YCUBUd-4BQC%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-cundy-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-cundy_71-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCundy1956" class="citation journal cs1">Cundy, H. Martyn (1956). "2642. 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Springer, Berlin. pp. <span class="nowrap">280–</span>291. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-71133-9">10.1007/978-3-540-71133-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-71132-2" title="Special:BookSources/978-3-540-71132-2"><bdi>978-3-540-71132-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2335496">2335496</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+16%3A+Volume+of+Polytopes+and+Hilbert%27s+Third+Problem&rft.btitle=Convex+and+Discrete+Geometry&rft.series=Grundlehren+der+mathematischen+Wissenschaften+%5BFundamental+Principles+of+Mathematical+Sciences%5D&rft.pages=%3Cspan+class%3D%22nowrap%22%3E280-%3C%2Fspan%3E291&rft.pub=Springer%2C+Berlin&rft.date=2007&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2335496%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-540-71133-9&rft.isbn=978-3-540-71132-2&rft.aulast=Gruber&rft.aufirst=Peter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-zeeman-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-zeeman_73-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZeeman2002" class="citation journal cs1"><a href="/wiki/Christopher_Zeeman" title="Christopher Zeeman">Zeeman, E. 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"On Hilbert's third problem". <i><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></i>. <b>86</b> (506): <span class="nowrap">241–</span>247. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3621846">10.2307/3621846</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3621846">3621846</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=On+Hilbert%27s+third+problem&rft.volume=86&rft.issue=506&rft.pages=%3Cspan+class%3D%22nowrap%22%3E241-%3C%2Fspan%3E247&rft.date=2002-07&rft_id=info%3Adoi%2F10.2307%2F3621846&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3621846%23id-name%3DJSTOR&rft.aulast=Zeeman&rft.aufirst=E.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-lm-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-lm_74-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLagariasMoews1995" class="citation journal cs1"><a href="/wiki/Jeffrey_Lagarias" title="Jeffrey Lagarias">Lagarias, J. C.</a>; Moews, D. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02574064">"Polytopes that fill <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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" /></span> and scissors congruence"</a>. <i><a href="/wiki/Discrete_%26_Computational_Geometry" title="Discrete & Computational Geometry">Discrete & Computational Geometry</a></i>. <b>13</b> (<span class="nowrap">3–</span>4): <span class="nowrap">573–</span>583. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02574064">10.1007/BF02574064</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1318797">1318797</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+%26+Computational+Geometry&rft.atitle=Polytopes+that+fill+MATH+RENDER+ERROR+and+scissors+congruence&rft.volume=13&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E573-%3C%2Fspan%3E583&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2FBF02574064&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1318797%23id-name%3DMR&rft.aulast=Lagarias&rft.aufirst=J.+C.&rft.au=Moews%2C+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02574064&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-erdahl-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-erdahl_75-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFErdahl1999" class="citation journal cs1">Erdahl, R. M. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Feujc.1999.0294">"Zonotopes, dicings, and Voronoi's conjecture on parallelohedra"</a>. <i>European Journal of Combinatorics</i>. <b>20</b> (6): <span class="nowrap">527–</span>549. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Feujc.1999.0294">10.1006/eujc.1999.0294</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1703597">1703597</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Combinatorics&rft.atitle=Zonotopes%2C+dicings%2C+and+Voronoi%27s+conjecture+on+parallelohedra&rft.volume=20&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E527-%3C%2Fspan%3E549&rft.date=1999&rft_id=info%3Adoi%2F10.1006%2Feujc.1999.0294&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1703597%23id-name%3DMR&rft.aulast=Erdahl&rft.aufirst=R.+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Feujc.1999.0294&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span>. Voronoi conjectured that all tilings of higher dimensional spaces by translates of a single <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytope</a> are combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case of <a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">zonotopes</a>. But as he writes (p. 429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGrünbaumShephard1980" class="citation journal cs1"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a>; <a href="/wiki/Geoffrey_Colin_Shephard" title="Geoffrey Colin Shephard">Shephard, G. C.</a> (1980). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1980-14827-2">"Tilings with congruent tiles"</a>. <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>. New Series. <b>3</b> (3): <span class="nowrap">951–</span>973. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1980-14827-2">10.1090/S0273-0979-1980-14827-2</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0585178">0585178</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Tilings+with+congruent+tiles&rft.volume=3&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E951-%3C%2Fspan%3E973&rft.date=1980&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-1980-14827-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D585178%23id-name%3DMR&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rft.au=Shephard%2C+G.+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0273-0979-1980-14827-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-alexandrov-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-alexandrov_76-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAlexandrov2005" class="citation book cs1"><a href="/wiki/Aleksandr_Danilovich_Aleksandrov" class="mw-redirect" title="Aleksandr Danilovich Aleksandrov">Alexandrov, A. D.</a> (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R9vPatr5aqYC&pg=PA349">"8.1 Parallelohedra"</a>. <a href="/wiki/Convex_Polyhedra_(book)" title="Convex Polyhedra (book)"><i>Convex Polyhedra</i></a>. Springer. pp. <span class="nowrap">349–</span>359.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8.1+Parallelohedra&rft.btitle=Convex+Polyhedra&rft.pages=%3Cspan+class%3D%22nowrap%22%3E349-%3C%2Fspan%3E359&rft.pub=Springer&rft.date=2005&rft.aulast=Alexandrov&rft.aufirst=A.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR9vPatr5aqYC%26pg%3DPA349&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-shephard-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-shephard_77-0">^</a></b></span> <span class="reference-text">In higher dimensions, however, there exist parallelopes that are not zonotopes. See e.g. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShephard1974" class="citation journal cs1">Shephard, G. C. (1974). "Space-filling zonotopes". <i>Mathematika</i>. <b>21</b> (2): <span class="nowrap">261–</span>269. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2FS0025579300008652">10.1112/S0025579300008652</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0365332">0365332</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematika&rft.atitle=Space-filling+zonotopes&rft.volume=21&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E261-%3C%2Fspan%3E269&rft.date=1974&rft_id=info%3Adoi%2F10.1112%2FS0025579300008652&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D365332%23id-name%3DMR&rft.aulast=Shephard&rft.aufirst=G.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-twelveessay-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-twelveessay_78-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoxeter1968" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1968). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p4o-Uf-i-IUC&pg=PA212"><i>The Beauty of Geometry: Twelve Essays</i></a>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. p. 167. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-40919-1" title="Special:BookSources/978-0-486-40919-1"><bdi>978-0-486-40919-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Beauty+of+Geometry%3A+Twelve+Essays&rft.pages=167&rft.pub=Dover+Publications&rft.date=1968&rft.isbn=978-0-486-40919-1&rft.aulast=Coxeter&rft.aufirst=H.+S.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dp4o-Uf-i-IUC%26pg%3DPA212&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span> See table III.</span> </li> <li id="cite_note-ns-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-ns_79-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNelsonSegerman2017" class="citation journal cs1">Nelson, Roice; Segerman, Henry (2017). "Visualizing hyperbolic honeycombs". <i>Journal of Mathematics and the Arts</i>. <b>11</b> (1): <span class="nowrap">4–</span>39. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1511.02851">1511.02851</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F17513472.2016.1263789">10.1080/17513472.2016.1263789</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematics+and+the+Arts&rft.atitle=Visualizing+hyperbolic+honeycombs&rft.volume=11&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4-%3C%2Fspan%3E39&rft.date=2017&rft_id=info%3Aarxiv%2F1511.02851&rft_id=info%3Adoi%2F10.1080%2F17513472.2016.1263789&rft.aulast=Nelson&rft.aufirst=Roice&rft.au=Segerman%2C+Henry&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-lunnon-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-lunnon_80-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLunnon1972" class="citation book cs1">Lunnon, W. F. (1972). "Symmetry of Cubical and General Polyominoes". In Read, Ronald C. (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ja7iBQAAQBAJ&pg=PA101"><i>Graph Theory and Computing</i></a>. New York: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. pp. <span class="nowrap">101–</span>108. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-48325-512-5" title="Special:BookSources/978-1-48325-512-5"><bdi>978-1-48325-512-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Symmetry+of+Cubical+and+General+Polyominoes&rft.btitle=Graph+Theory+and+Computing&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E101-%3C%2Fspan%3E108&rft.pub=Academic+Press&rft.date=1972&rft.isbn=978-1-48325-512-5&rft.aulast=Lunnon&rft.aufirst=W.+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dja7iBQAAQBAJ%26pg%3DPA101&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-hut-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-hut_81-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDiazO'Rourke2015" class="citation arxiv cs1">Diaz, Giovanna; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (2015). "Hypercube unfoldings that tile <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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" /></span> and <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" /></span>". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1512.02086">1512.02086</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CG">cs.CG</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Hypercube+unfoldings+that+tile+MATH+RENDER+ERROR+and+MATH+RENDER+ERROR&rft.date=2015&rft_id=info%3Aarxiv%2F1512.02086&rft.aulast=Diaz&rft.aufirst=Giovanna&rft.au=O%27Rourke%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-pucc-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-pucc_82-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLangermanWinslow2016" class="citation conference cs1"><a href="/wiki/Stefan_Langerman" title="Stefan Langerman">Langerman, Stefan</a>; Winslow, Andrew (2016). <a rel="nofollow" class="external text" href="http://andrewwinslow.com/papers/polyunfold-jcdcggg16.pdf">"Polycube unfoldings satisfying Conway's criterion"</a> <span class="cs1-format">(PDF)</span>. <i>19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016)</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Polycube+unfoldings+satisfying+Conway%27s+criterion&rft.btitle=19th+Japan+Conference+on+Discrete+and+Computational+Geometry%2C+Graphs%2C+and+Games+%28JCDCG%5E3+2016%29&rft.date=2016&rft.aulast=Langerman&rft.aufirst=Stefan&rft.au=Winslow%2C+Andrew&rft_id=http%3A%2F%2Fandrewwinslow.com%2Fpapers%2Fpolyunfold-jcdcggg16.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-hall-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-hall_83-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHall1893" class="citation journal cs1"><a href="/wiki/T._Proctor_Hall" title="T. 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In Goldstein, Susan; McKenna, Douglas; Fenyvesi, Kristóf (eds.). <i>Bridges 2019 Conference Proceedings</i>. <a href="/wiki/Linz,_Austria" class="mw-redirect" title="Linz, Austria">Linz, Austria</a>: Tessellations Publishing, <a href="/wiki/Phoenix,_Arizona" title="Phoenix, Arizona">Phoenix, Arizona</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-938664-30-4" title="Special:BookSources/978-1-938664-30-4"><bdi>978-1-938664-30-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Max+Br%C3%BCckner%CA%BCs+Wunderkammer+of+Paper+Polyhedra&rft.btitle=Bridges+2019+Conference+Proceedings&rft.place=Linz%2C+Austria&rft.pub=Tessellations+Publishing%2C+Phoenix%2C+Arizona&rft.date=2019-07-16%2F2019-07-20&rft.isbn=978-1-938664-30-4&rft.aulast=Hart&rft.aufirst=George&rft_id=https%3A%2F%2Farchive.bridgesmathart.org%2F2019%2Fbridges2019-59.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-popko-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-popko_86-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPopko2012" class="citation book cs1">Popko, Edward S. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WLAFlr1_2S4C&pg=PA100"><i>Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere</i></a>. CRC Press. pp. <span class="nowrap">100–</span>101. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781466504295" title="Special:BookSources/9781466504295"><bdi>9781466504295</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Divided+Spheres%3A+Geodesics+and+the+Orderly+Subdivision+of+the+Sphere&rft.pages=%3Cspan+class%3D%22nowrap%22%3E100-%3C%2Fspan%3E101&rft.pub=CRC+Press&rft.date=2012&rft.isbn=9781466504295&rft.aulast=Popko&rft.aufirst=Edward+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWLAFlr1_2S4C%26pg%3DPA100&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-fuller-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-fuller_87-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFuller1975" class="citation book cs1">Fuller, Buckimster (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AKDgDQAAQBAJ&pg=PA173"><i>Synergetics: Explorations in the Geometry of Thinking</i></a>. MacMillan Publishing. p. 173. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-065320-2" title="Special:BookSources/978-0-02-065320-2"><bdi>978-0-02-065320-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Synergetics%3A+Explorations+in+the+Geometry+of+Thinking&rft.pages=173&rft.pub=MacMillan+Publishing&rft.date=1975&rft.isbn=978-0-02-065320-2&rft.aulast=Fuller&rft.aufirst=Buckimster&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAKDgDQAAQBAJ%26pg%3DPA173&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-yackel-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-yackel_88-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYackel2013" class="citation conference cs1">Yackel, Carolyn (26–30 July 2013). <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2009/bridges2009-123.pdf">"Marking a Physical Sphere with a Projected Platonic Solid"</a> <span class="cs1-format">(PDF)</span>. In <a href="/wiki/Craig_S._Kaplan" title="Craig S. Kaplan">Kaplan, Craig</a>; Sarhangi, Reza (eds.). <i>Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture</i>. <a href="/wiki/Banff,_Alberta" title="Banff, Alberta">Banff, Alberta</a>, Canada. pp. <span class="nowrap">123–</span>130. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-96652-020-0" title="Special:BookSources/978-0-96652-020-0"><bdi>978-0-96652-020-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Marking+a+Physical+Sphere+with+a+Projected+Platonic+Solid&rft.btitle=Proceedings+of+Bridges+2009%3A+Mathematics%2C+Music%2C+Art%2C+Architecture%2C+Culture&rft.place=Banff%2C+Alberta%2C+Canada&rft.pages=%3Cspan+class%3D%22nowrap%22%3E123-%3C%2Fspan%3E130&rft.date=2013-07-26%2F2013-07-30&rft.isbn=978-0-96652-020-0&rft.aulast=Yackel&rft.aufirst=Carolyn&rft_id=https%3A%2F%2Farchive.bridgesmathart.org%2F2009%2Fbridges2009-123.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> <li id="cite_note-marar-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-marar_89-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMarat2022" class="citation book cs1">Marat, Ton (2022). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aPGGEAAAQBAJ&pg=PA112"><i>A Ludic Journey into Geometric Topology</i></a>. Springer. p. 112. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-031-07442-4">10.1007/978-3-031-07442-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-07442-4" title="Special:BookSources/978-3-031-07442-4"><bdi>978-3-031-07442-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Ludic+Journey+into+Geometric+Topology&rft.pages=112&rft.pub=Springer&rft.date=2022&rft_id=info%3Adoi%2F10.1007%2F978-3-031-07442-4&rft.isbn=978-3-031-07442-4&rft.aulast=Marat&rft.aufirst=Ton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaPGGEAAAQBAJ%26pg%3DPA112&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size notpageimage" typeof="mw:File/Frameless"><a href="/wiki/File:Stereographic_projection_in_3D.svg" class="mw-file-description"><img alt="Stereographic projection from the top of a sphere onto a plane beneath it" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/220px-Stereographic_projection_in_3D.svg.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/330px-Stereographic_projection_in_3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/440px-Stereographic_projection_in_3D.svg.png 2x" data-file-width="870" data-file-height="639" /></a></span><div class="sidebar-caption"><a href="/wiki/Projective_geometry" title="Projective geometry">Projecting</a> a <a href="/wiki/Sphere" title="Sphere">sphere</a> to a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a> <ul><li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li> <li><a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a> <ul><li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a> <ul><li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a> <ul><li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital</a></li></ul></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence </a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a> <ul><li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="/wiki/Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a> <ul><li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a></li> <li><a href="/wiki/Curve" title="Curve">Curve</a></li> <li><a href="/wiki/Diagonal" title="Diagonal">Diagonal</a></li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a>)</li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Line_(geometry)" title="Line (geometry)">Line</a> <ul><li><a href="/wiki/Line_segment" title="Line segment">segment</a></li> <li><a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li> <li><a href="/wiki/Length" title="Length">Length</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li> <li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li> <li><a href="/wiki/Hypotenuse" title="Hypotenuse">Hypotenuse</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Diameter" title="Diameter">Diameter</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Volume" title="Volume">Volume</a></li></ul> <ul><li><a class="mw-selflink selflink">Cube</a> <ul><li><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></li></ul></li> <li><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li> <li><a href="/wiki/Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a>-/other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li> <li><a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;"> <a href="/wiki/List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1–1400s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1400s–1700s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1700s–1900s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Present day</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Cube.html">"Cube"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Cube&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCube.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACube" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071009235233/http://polyhedra.org/poly/show/1/cube">Cube: Interactive Polyhedron Model</a>*</li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/cubevolume.html">Volume of a cube</a>, with interactive animation</li> <li><a rel="nofollow" class="external text" href="http://www.software3d.com/Cube.php">Cube</a> (Robert Webb's site)</li></ul> <div class="navbox-styles"><link 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.navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><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_polyhedra686" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed 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_polyhedra686" 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 class="mw-selflink selflink">cube</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">octahedron</a></li> <li><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a></li> <li><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a><br /><span class="nobold">(<a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li> <li><a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li> <li><a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a></li> <li><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li> <li><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li> <li><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li> <li><a href="/wiki/Snub_cube" title="Snub cube">snub cube</a></li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li> <li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li> <li><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li> <li><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a><br /><span class="nobold">(duals of Archimedean)</span></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li> <li><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li> <li><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li> <li><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li> <li><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li> <li><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li> <li><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li> <li><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Dihedron" title="Dihedron">dihedron</a></i></li> <li><i><a href="/wiki/Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a></li> <li><a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral others</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></li> <li><a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedra</a></li> <li><a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramid</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Bicupola_(geometry)" class="mw-redirect" title="Bicupola (geometry)">bicupola</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Bifrustum" title="Bifrustum">bifrustum</a></li> <li><a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotunda</a></li> <li><a href="/wiki/Birotunda" title="Birotunda">birotunda</a></li> <li><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a></li> <li><a href="/wiki/Scutoid" title="Scutoid">scutoid</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Degenerate polyhedra are in <i>italics</i>.</div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Catalan_solids18" 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:Catalan_solids" title="Template:Catalan solids"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Catalan_solids" title="Template talk:Catalan solids"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Catalan_solids" title="Special:EditPage/Template:Catalan solids"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Catalan_solids18" style="font-size:114%;margin:0 4em"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0;padding:0"><div style="padding:0 0.25em"> <table style="width:100%"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_4a_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Polyhedron_truncated_4a_dual.png/90px-Polyhedron_truncated_4a_dual.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Polyhedron_truncated_4a_dual.png/135px-Polyhedron_truncated_4a_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Polyhedron_truncated_4a_dual.png/180px-Polyhedron_truncated_4a_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">Triakis tetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Needle)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_4b_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Polyhedron_truncated_4b_dual.png/90px-Polyhedron_truncated_4b_dual.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Polyhedron_truncated_4b_dual.png/135px-Polyhedron_truncated_4b_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Polyhedron_truncated_4b_dual.png/180px-Polyhedron_truncated_4b_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">Triakis tetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Kis)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_6_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Polyhedron_truncated_6_dual.png/90px-Polyhedron_truncated_6_dual.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Polyhedron_truncated_6_dual.png/135px-Polyhedron_truncated_6_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Polyhedron_truncated_6_dual.png/180px-Polyhedron_truncated_6_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">Triakis octahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Needle)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_8_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Polyhedron_truncated_8_dual.png/90px-Polyhedron_truncated_8_dual.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Polyhedron_truncated_8_dual.png/135px-Polyhedron_truncated_8_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Polyhedron_truncated_8_dual.png/180px-Polyhedron_truncated_8_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">Tetrakis hexahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Kis)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_12_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Polyhedron_truncated_12_dual.png/90px-Polyhedron_truncated_12_dual.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Polyhedron_truncated_12_dual.png/135px-Polyhedron_truncated_12_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Polyhedron_truncated_12_dual.png/180px-Polyhedron_truncated_12_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">Triakis icosahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Needle)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_20_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Polyhedron_truncated_20_dual.png/90px-Polyhedron_truncated_20_dual.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Polyhedron_truncated_20_dual.png/135px-Polyhedron_truncated_20_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Polyhedron_truncated_20_dual.png/180px-Polyhedron_truncated_20_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">Pentakis dodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Kis)</a> </td></tr> <tr> <td colspan="2" style="opacity: .4;"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_4-4_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Polyhedron_4-4_dual.png/250px-Polyhedron_4-4_dual.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Polyhedron_4-4_dual.png/330px-Polyhedron_4-4_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Polyhedron_4-4_dual.png/500px-Polyhedron_4-4_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><big><a class="mw-selflink selflink">Rhombic hexahedron</a></big><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Join)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_6-8_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Polyhedron_6-8_dual.png/180px-Polyhedron_6-8_dual.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Polyhedron_6-8_dual.png/270px-Polyhedron_6-8_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Polyhedron_6-8_dual.png/360px-Polyhedron_6-8_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><big><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Rhombic dodecahedron</a></big><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Join)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_12-20_dual.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Polyhedron_12-20_dual.png/180px-Polyhedron_12-20_dual.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Polyhedron_12-20_dual.png/270px-Polyhedron_12-20_dual.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Polyhedron_12-20_dual.png/360px-Polyhedron_12-20_dual.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><big><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">Rhombic triacontahedron</a></big><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Join)</a> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_small_rhombi_4-4_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Polyhedron_small_rhombi_4-4_dual_max.png/90px-Polyhedron_small_rhombi_4-4_dual_max.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Polyhedron_small_rhombi_4-4_dual_max.png/135px-Polyhedron_small_rhombi_4-4_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Polyhedron_small_rhombi_4-4_dual_max.png/180px-Polyhedron_small_rhombi_4-4_dual_max.png 2x" data-file-width="3872" data-file-height="3927" /></a></span><br /><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Deltoidal dodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Ortho)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_great_rhombi_4-4_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Polyhedron_great_rhombi_4-4_dual_max.png/90px-Polyhedron_great_rhombi_4-4_dual_max.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Polyhedron_great_rhombi_4-4_dual_max.png/135px-Polyhedron_great_rhombi_4-4_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Polyhedron_great_rhombi_4-4_dual_max.png/180px-Polyhedron_great_rhombi_4-4_dual_max.png 2x" data-file-width="3861" data-file-height="3916" /></a></span><br /><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">Disdyakis hexahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Meta)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_small_rhombi_6-8_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Polyhedron_small_rhombi_6-8_dual_max.png/90px-Polyhedron_small_rhombi_6-8_dual_max.png" decoding="async" width="90" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Polyhedron_small_rhombi_6-8_dual_max.png/135px-Polyhedron_small_rhombi_6-8_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Polyhedron_small_rhombi_6-8_dual_max.png/180px-Polyhedron_small_rhombi_6-8_dual_max.png 2x" data-file-width="3850" data-file-height="3923" /></a></span><br /><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">Deltoidal icositetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Ortho)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_great_rhombi_6-8_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Polyhedron_great_rhombi_6-8_dual_max.png/90px-Polyhedron_great_rhombi_6-8_dual_max.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Polyhedron_great_rhombi_6-8_dual_max.png/135px-Polyhedron_great_rhombi_6-8_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Polyhedron_great_rhombi_6-8_dual_max.png/180px-Polyhedron_great_rhombi_6-8_dual_max.png 2x" data-file-width="3878" data-file-height="3928" /></a></span><br /><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">Disdyakis dodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Meta)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_small_rhombi_12-20_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Polyhedron_small_rhombi_12-20_dual_max.png/90px-Polyhedron_small_rhombi_12-20_dual_max.png" decoding="async" width="90" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Polyhedron_small_rhombi_12-20_dual_max.png/135px-Polyhedron_small_rhombi_12-20_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Polyhedron_small_rhombi_12-20_dual_max.png/180px-Polyhedron_small_rhombi_12-20_dual_max.png 2x" data-file-width="3954" data-file-height="3911" /></a></span><br /><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">Deltoidal hexecontahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Ortho)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_great_rhombi_12-20_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Polyhedron_great_rhombi_12-20_dual_max.png/90px-Polyhedron_great_rhombi_12-20_dual_max.png" decoding="async" width="90" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Polyhedron_great_rhombi_12-20_dual_max.png/135px-Polyhedron_great_rhombi_12-20_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Polyhedron_great_rhombi_12-20_dual_max.png/180px-Polyhedron_great_rhombi_12-20_dual_max.png 2x" data-file-width="3928" data-file-height="3773" /></a></span><br /><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">Disdyakis triacontahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Meta)</a> </td></tr> <tr> <td colspan="2" style="opacity: .4;"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_4-4_right_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Polyhedron_snub_4-4_right_dual_max.png/82px-Polyhedron_snub_4-4_right_dual_max.png" decoding="async" width="82" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Polyhedron_snub_4-4_right_dual_max.png/123px-Polyhedron_snub_4-4_right_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Polyhedron_snub_4-4_right_dual_max.png/164px-Polyhedron_snub_4-4_right_dual_max.png 2x" data-file-width="3982" data-file-height="3757" /></a></span><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_4-4_left_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Polyhedron_snub_4-4_left_dual_max.png/80px-Polyhedron_snub_4-4_left_dual_max.png" decoding="async" width="80" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Polyhedron_snub_4-4_left_dual_max.png/120px-Polyhedron_snub_4-4_left_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Polyhedron_snub_4-4_left_dual_max.png/160px-Polyhedron_snub_4-4_left_dual_max.png 2x" data-file-width="3652" data-file-height="3960" /></a></span><br /><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Pentagonal dodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Gyro)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_6-8_right_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Polyhedron_snub_6-8_right_dual_max.png/80px-Polyhedron_snub_6-8_right_dual_max.png" decoding="async" width="80" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Polyhedron_snub_6-8_right_dual_max.png/120px-Polyhedron_snub_6-8_right_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Polyhedron_snub_6-8_right_dual_max.png/160px-Polyhedron_snub_6-8_right_dual_max.png 2x" data-file-width="3833" data-file-height="3889" /></a></span><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_6-8_left_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Polyhedron_snub_6-8_left_dual_max.png/120px-Polyhedron_snub_6-8_left_dual_max.png" decoding="async" width="80" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Polyhedron_snub_6-8_left_dual_max.png/250px-Polyhedron_snub_6-8_left_dual_max.png 2x" data-file-width="3833" data-file-height="3895" /></a></span><br /><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">Pentagonal icositetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Gyro)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_12-20_right_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Polyhedron_snub_12-20_right_dual_max.png/80px-Polyhedron_snub_12-20_right_dual_max.png" decoding="async" width="80" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Polyhedron_snub_12-20_right_dual_max.png/120px-Polyhedron_snub_12-20_right_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Polyhedron_snub_12-20_right_dual_max.png/160px-Polyhedron_snub_12-20_right_dual_max.png 2x" data-file-width="3928" data-file-height="3893" /></a></span><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_12-20_left_dual_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Polyhedron_snub_12-20_left_dual_max.png/80px-Polyhedron_snub_12-20_left_dual_max.png" decoding="async" width="80" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Polyhedron_snub_12-20_left_dual_max.png/120px-Polyhedron_snub_12-20_left_dual_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Polyhedron_snub_12-20_left_dual_max.png/160px-Polyhedron_snub_12-20_left_dual_max.png 2x" data-file-width="3939" data-file-height="3877" /></a></span><br /><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">Pentagonal hexecontahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Gyro)</a> </td></tr> <tr> <td colspan="8"> <div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Archimedean_duals39" 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"><div id="Archimedean_duals39" style="font-size:114%;margin:0 4em"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean</a> duals</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0;padding:0"><div style="padding:0 0.25em"> <table style="width:100%"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_4a.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Polyhedron_truncated_4a.png/90px-Polyhedron_truncated_4a.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Polyhedron_truncated_4a.png/135px-Polyhedron_truncated_4a.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Polyhedron_truncated_4a.png/180px-Polyhedron_truncated_4a.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">Truncated tetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Truncate)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_4b.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Polyhedron_truncated_4b.png/90px-Polyhedron_truncated_4b.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Polyhedron_truncated_4b.png/135px-Polyhedron_truncated_4b.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Polyhedron_truncated_4b.png/180px-Polyhedron_truncated_4b.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">Truncated tetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Zip)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Polyhedron_truncated_6.png/90px-Polyhedron_truncated_6.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Polyhedron_truncated_6.png/135px-Polyhedron_truncated_6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Polyhedron_truncated_6.png/180px-Polyhedron_truncated_6.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Truncated_cube" title="Truncated cube">Truncated cube</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Truncate)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Polyhedron_truncated_8.png/90px-Polyhedron_truncated_8.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Polyhedron_truncated_8.png/135px-Polyhedron_truncated_8.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Polyhedron_truncated_8.png/180px-Polyhedron_truncated_8.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">Truncated octahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Zip)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_12.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Polyhedron_truncated_12.png/90px-Polyhedron_truncated_12.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Polyhedron_truncated_12.png/135px-Polyhedron_truncated_12.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Polyhedron_truncated_12.png/180px-Polyhedron_truncated_12.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">Truncated dodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Truncate)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_truncated_20.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Polyhedron_truncated_20.png/90px-Polyhedron_truncated_20.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Polyhedron_truncated_20.png/135px-Polyhedron_truncated_20.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Polyhedron_truncated_20.png/180px-Polyhedron_truncated_20.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">Truncated icosahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Zip)</a> </td></tr> <tr> <td colspan="2" style="opacity: .4;"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_4-4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Polyhedron_4-4.png/180px-Polyhedron_4-4.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Polyhedron_4-4.png/270px-Polyhedron_4-4.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Polyhedron_4-4.png/360px-Polyhedron_4-4.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><big><a href="/wiki/Octahedron" title="Octahedron">Tetratetrahedron</a></big><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Ambo)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_6-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Polyhedron_6-8.png/180px-Polyhedron_6-8.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Polyhedron_6-8.png/270px-Polyhedron_6-8.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Polyhedron_6-8.png/360px-Polyhedron_6-8.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><big><a href="/wiki/Cuboctahedron" title="Cuboctahedron">Cuboctahedron</a></big><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Ambo)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_12-20.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Polyhedron_12-20.png/180px-Polyhedron_12-20.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Polyhedron_12-20.png/270px-Polyhedron_12-20.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Polyhedron_12-20.png/360px-Polyhedron_12-20.png 2x" data-file-width="4000" data-file-height="4000" /></a></span><br /><big><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a></big><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Ambo)</a> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_small_rhombi_4-4_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Polyhedron_small_rhombi_4-4_max.png/90px-Polyhedron_small_rhombi_4-4_max.png" decoding="async" width="90" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Polyhedron_small_rhombi_4-4_max.png/135px-Polyhedron_small_rhombi_4-4_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Polyhedron_small_rhombi_4-4_max.png/180px-Polyhedron_small_rhombi_4-4_max.png 2x" data-file-width="3850" data-file-height="3680" /></a></span><br /><a href="/wiki/Cuboctahedron" title="Cuboctahedron">Rhombitetratetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Expand)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_great_rhombi_4-4_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Polyhedron_great_rhombi_4-4_max.png/90px-Polyhedron_great_rhombi_4-4_max.png" decoding="async" width="90" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Polyhedron_great_rhombi_4-4_max.png/135px-Polyhedron_great_rhombi_4-4_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Polyhedron_great_rhombi_4-4_max.png/180px-Polyhedron_great_rhombi_4-4_max.png 2x" data-file-width="3922" data-file-height="3854" /></a></span><br /><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">Truncated tetratetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Bevel)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_small_rhombi_6-8_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Polyhedron_small_rhombi_6-8_max.png/90px-Polyhedron_small_rhombi_6-8_max.png" decoding="async" width="90" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Polyhedron_small_rhombi_6-8_max.png/135px-Polyhedron_small_rhombi_6-8_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Polyhedron_small_rhombi_6-8_max.png/180px-Polyhedron_small_rhombi_6-8_max.png 2x" data-file-width="3906" data-file-height="3910" /></a></span><br /><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">Rhombicuboctahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Expand)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_great_rhombi_6-8_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Polyhedron_great_rhombi_6-8_max.png/90px-Polyhedron_great_rhombi_6-8_max.png" decoding="async" width="90" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Polyhedron_great_rhombi_6-8_max.png/135px-Polyhedron_great_rhombi_6-8_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Polyhedron_great_rhombi_6-8_max.png/180px-Polyhedron_great_rhombi_6-8_max.png 2x" data-file-width="3860" data-file-height="3795" /></a></span><br /><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">Truncated cuboctahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Bevel)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_small_rhombi_12-20_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Polyhedron_small_rhombi_12-20_max.png/90px-Polyhedron_small_rhombi_12-20_max.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Polyhedron_small_rhombi_12-20_max.png/135px-Polyhedron_small_rhombi_12-20_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Polyhedron_small_rhombi_12-20_max.png/180px-Polyhedron_small_rhombi_12-20_max.png 2x" data-file-width="3973" data-file-height="4000" /></a></span><br /><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">Rhombicosidodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Expand)</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyhedron_great_rhombi_12-20_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Polyhedron_great_rhombi_12-20_max.png/90px-Polyhedron_great_rhombi_12-20_max.png" decoding="async" width="90" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Polyhedron_great_rhombi_12-20_max.png/135px-Polyhedron_great_rhombi_12-20_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Polyhedron_great_rhombi_12-20_max.png/180px-Polyhedron_great_rhombi_12-20_max.png 2x" data-file-width="3943" data-file-height="3977" /></a></span><br /><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">Truncated icosidodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Bevel)</a> </td></tr> <tr> <td colspan="2" style="opacity: .4;"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_4-4_left_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Polyhedron_snub_4-4_left_max.png/88px-Polyhedron_snub_4-4_left_max.png" decoding="async" width="88" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Polyhedron_snub_4-4_left_max.png/132px-Polyhedron_snub_4-4_left_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Polyhedron_snub_4-4_left_max.png/176px-Polyhedron_snub_4-4_left_max.png 2x" data-file-width="3926" data-file-height="3526" /></a></span><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_4-4_right_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Polyhedron_snub_4-4_right_max.png/80px-Polyhedron_snub_4-4_right_max.png" decoding="async" width="80" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Polyhedron_snub_4-4_right_max.png/120px-Polyhedron_snub_4-4_right_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Polyhedron_snub_4-4_right_max.png/160px-Polyhedron_snub_4-4_right_max.png 2x" data-file-width="3408" data-file-height="3829" /></a></span><br /><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">Snub tetrahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Snub)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_6-8_left_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Polyhedron_snub_6-8_left_max.png/80px-Polyhedron_snub_6-8_left_max.png" decoding="async" width="80" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Polyhedron_snub_6-8_left_max.png/120px-Polyhedron_snub_6-8_left_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Polyhedron_snub_6-8_left_max.png/160px-Polyhedron_snub_6-8_left_max.png 2x" data-file-width="3932" data-file-height="3804" /></a></span><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_6-8_right_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Polyhedron_snub_6-8_right_max.png/80px-Polyhedron_snub_6-8_right_max.png" decoding="async" width="80" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Polyhedron_snub_6-8_right_max.png/120px-Polyhedron_snub_6-8_right_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Polyhedron_snub_6-8_right_max.png/160px-Polyhedron_snub_6-8_right_max.png 2x" data-file-width="3905" data-file-height="3860" /></a></span><br /><a href="/wiki/Snub_cube" title="Snub cube">Snub cube</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Snub)</a> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_12-20_left_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Polyhedron_snub_12-20_left_max.png/80px-Polyhedron_snub_12-20_left_max.png" decoding="async" width="80" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Polyhedron_snub_12-20_left_max.png/120px-Polyhedron_snub_12-20_left_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Polyhedron_snub_12-20_left_max.png/160px-Polyhedron_snub_12-20_left_max.png 2x" data-file-width="3966" data-file-height="4000" /></a></span><span typeof="mw:File"><a href="/wiki/File:Polyhedron_snub_12-20_right_max.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Polyhedron_snub_12-20_right_max.png/80px-Polyhedron_snub_12-20_right_max.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Polyhedron_snub_12-20_right_max.png/120px-Polyhedron_snub_12-20_right_max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Polyhedron_snub_12-20_right_max.png/160px-Polyhedron_snub_12-20_right_max.png 2x" data-file-width="3983" data-file-height="4000" /></a></span><br /><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">Snub dodecahedron</a><br /><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">(Snub)</a> </td></tr></tbody></table> </div></td></tr></tbody></table></div> </td></tr></tbody></table> </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:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox1489" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q812880#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4079396-5">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Cube"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85034644">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Cube"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11947058p">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Cube"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11947058p">BnF data</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="krychle"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph1254474&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007535952905171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" 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<script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-858ccf4554-zdtf9","wgBackendResponseTime":257,"wgPageParseReport":{"limitreport":{"cputime":"2.010","walltime":"2.331","ppvisitednodes":{"value":26605,"limit":1000000},"postexpandincludesize":{"value":274355,"limit":2097152},"templateargumentsize":{"value":7747,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":16,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":392029,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1852.651 1 -total"," 40.50% 750.305 1 Template:Reflist"," 21.64% 400.997 47 Template:Cite_book"," 21.58% 399.823 78 Template:R"," 20.31% 376.241 88 Template:R/ref"," 10.08% 186.672 31 Template:Cite_journal"," 7.20% 133.370 5 Template:Sfnp"," 6.83% 126.598 88 Template:R/superscript"," 5.84% 108.173 1 Template:General_geometry"," 5.74% 106.385 88 Template:R/where"]},"scribunto":{"limitreport-timeusage":{"value":"1.083","limit":"10.000"},"limitreport-memusage":{"value":7879915,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAldousWilson2000\"] = 1,\n [\"CITEREFAlexandrov2005\"] = 1,\n [\"CITEREFAlsinaNelsen2015\"] = 1,\n [\"CITEREFBall2010\"] = 1,\n [\"CITEREFBarnes2012\"] = 1,\n [\"CITEREFBerman2014\"] = 1,\n [\"CITEREFBiegasiewiczGriffithsSavageTsanakstidis2015\"] = 1,\n [\"CITEREFCalterCalter2011\"] = 1,\n [\"CITEREFChartrandZhang2012\"] = 1,\n [\"CITEREFChinMohamedShariffIshikawa2021\"] = 1,\n [\"CITEREFCoxeter1968\"] = 1,\n [\"CITEREFCoxeter1973\"] = 1,\n [\"CITEREFCromwell1997\"] = 1,\n [\"CITEREFCundy1956\"] = 1,\n [\"CITEREFCundyRollett1961\"] = 1,\n [\"CITEREFCunninghamPellicer2024\"] = 1,\n [\"CITEREFDiazO\u0026#039;Rourke2015\"] = 1,\n [\"CITEREFErdahl1999\"] = 1,\n [\"CITEREFErickson2011\"] = 1,\n [\"CITEREFFowler2010\"] = 1,\n [\"CITEREFFrench1988\"] = 1,\n [\"CITEREFFuller1975\"] = 1,\n [\"CITEREFGreene1986\"] = 1,\n [\"CITEREFGruber2007\"] = 1,\n [\"CITEREFGrünbaum1997\"] = 1,\n [\"CITEREFGrünbaum2003\"] = 1,\n [\"CITEREFGrünbaumShephard1980\"] = 1,\n [\"CITEREFHaeckel1904\"] = 1,\n [\"CITEREFHall1893\"] = 1,\n [\"CITEREFHararyHayesWu1988\"] = 1,\n [\"CITEREFHart2019\"] = 1,\n [\"CITEREFHasanHossainLópez-OrtizNusrat2010\"] = 1,\n [\"CITEREFHeath1908\"] = 1,\n [\"CITEREFHelvajianJanson2008\"] = 1,\n [\"CITEREFHerrmannSally2013\"] = 1,\n [\"CITEREFHoffmann2020\"] = 1,\n [\"CITEREFHolme2010\"] = 1,\n [\"CITEREFHorvatPisanski2010\"] = 1,\n [\"CITEREFInchbald2006\"] = 1,\n [\"CITEREFJeon2009\"] = 1,\n [\"CITEREFJerrardWetzelYuan2017\"] = 1,\n [\"CITEREFJessen1967\"] = 1,\n [\"CITEREFJohn_D._Barrow1999\"] = 1,\n [\"CITEREFJohnson1966\"] = 1,\n [\"CITEREFJoyner2008\"] = 1,\n [\"CITEREFKemp1998\"] = 1,\n [\"CITEREFKhattar2008\"] = 1,\n [\"CITEREFKitaevLozin2015\"] = 1,\n [\"CITEREFKov´acsNagyStomfaiTurgay2021\"] = 1,\n [\"CITEREFKozachok2012\"] = 1,\n [\"CITEREFLagariasMoews1995\"] = 1,\n [\"CITEREFLangermanWinslow2016\"] = 1,\n [\"CITEREFLinti2013\"] = 1,\n [\"CITEREFLivio2003\"] = 1,\n [\"CITEREFLunnon1972\"] = 1,\n [\"CITEREFLützen2010\"] = 1,\n [\"CITEREFMarat2022\"] = 1,\n [\"CITEREFMarch1996\"] = 1,\n [\"CITEREFMcLean1990\"] = 1,\n [\"CITEREFMillsKolf1999\"] = 1,\n [\"CITEREFMoore2018\"] = 1,\n [\"CITEREFNelsonSegerman2017\"] = 1,\n [\"CITEREFPadmanabhan2015\"] = 1,\n [\"CITEREFPisanskiServatius2013\"] = 1,\n [\"CITEREFPoo-Sung2016\"] = 1,\n [\"CITEREFPopko2012\"] = 1,\n [\"CITEREFRajwade2001\"] = 1,\n [\"CITEREFReavenZeilten2006\"] = 1,\n [\"CITEREFRudolph2022\"] = 1,\n [\"CITEREFSenechal1989\"] = 1,\n [\"CITEREFShephard1974\"] = 1,\n [\"CITEREFSkilling1976\"] = 1,\n [\"CITEREFSlobodanObradovićÐukanović2015\"] = 1,\n [\"CITEREFSmith2000\"] = 1,\n [\"CITEREFSriraman2009\"] = 1,\n [\"CITEREFThomson1845\"] = 1,\n [\"CITEREFTimofeenko2010\"] = 1,\n [\"CITEREFTisza2001\"] = 1,\n [\"CITEREFVianaXavierAiresCampos2019\"] = 1,\n [\"CITEREFVollmerMöllmann2011\"] = 1,\n [\"CITEREFWalterDeloudi2009\"] = 1,\n [\"CITEREFYackel2013\"] = 1,\n [\"CITEREFZeeman2002\"] = 1,\n [\"CITEREFZiegler1995\"] = 1,\n [\"Compound_of_cubes\"] = 1,\n [\"Spherical_cube\"] = 1,\n}\ntemplate_list = table#1 {\n [\"-\"] = 2,\n [\"Anchor\"] = 2,\n [\"Authority control\"] = 1,\n [\"Catalan solids\"] = 1,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 47,\n [\"Cite conference\"] = 5,\n [\"Cite journal\"] = 31,\n [\"Convex polyhedron navigator\"] = 1,\n [\"General geometry\"] = 1,\n [\"Harvtxt\"] = 3,\n [\"Infobox polyhedron\"] = 1,\n [\"Main article\"] = 1,\n [\"Mathworld\"] = 1,\n [\"Multiple image\"] = 5,\n [\"Nowrap\"] = 1,\n [\"Other uses\"] = 1,\n [\"R\"] = 78,\n [\"Reflist\"] = 1,\n [\"Sfnp\"] = 5,\n [\"Short description\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n","limitreport-profile":[["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","220","18.6"],["recursiveClone \u003CmwInit.lua:45\u003E","100","8.5"],["?","80","6.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::match","80","6.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","60","5.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getEntityStatements","60","5.1"],["dataWrapper \u003Cmw.lua:672\u003E","60","5.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::len","40","3.4"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::find","40","3.4"],["(for generator)","40","3.4"],["[others]","400","33.9"]]},"cachereport":{"origin":"mw-api-int.codfw.main-754764944-jhbkv","timestamp":"20250312084101","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Cube","url":"https:\/\/en.wikipedia.org\/wiki\/Cube","sameAs":"http:\/\/www.wikidata.org\/entity\/Q812880","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q812880","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-09-07T17:57:17Z","dateModified":"2025-03-12T08:40:47Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/b6\/Cube-h.svg","headline":"three-dimensional solid object bounded by six square sides"}</script> </body> </html>