Bipyramid - Wikipedia

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href="#Related_and_other_types_of_bipyramid"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Related and other types of bipyramid</span> </div> </a> <button aria-controls="toc-Related_and_other_types_of_bipyramid-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 and other types of bipyramid subsection</span> </button> <ul id="toc-Related_and_other_types_of_bipyramid-sublist" class="vector-toc-list"> <li id="toc-Concave_bipyramids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Concave_bipyramids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Concave bipyramids</span> </div> </a> <ul id="toc-Concave_bipyramids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymmetric_bipyramids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymmetric_bipyramids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Asymmetric bipyramids</span> </div> </a> <ul id="toc-Asymmetric_bipyramids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scalene_triangle_bipyramids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scalene_triangle_bipyramids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Scalene triangle bipyramids</span> </div> </a> <ul id="toc-Scalene_triangle_bipyramids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scalenohedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scalenohedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Scalenohedra</span> </div> </a> <ul id="toc-Scalenohedra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Star_bipyramids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Star_bipyramids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Star bipyramids</span> </div> </a> <ul id="toc-Star_bipyramids-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-4-polytopes_with_bipyramidal_cells" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#4-polytopes_with_bipyramidal_cells"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>4-polytopes with bipyramidal cells</span> </div> </a> <ul id="toc-4-polytopes_with_bipyramidal_cells-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other dimensions</span> </div> </a> <ul id="toc-Other_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_Cited" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Works_Cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Works Cited</span> </div> </a> <ul id="toc-Works_Cited-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">9</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" > 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Available in 27 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-27" 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">27 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87%D8%B1%D9%85_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A" title="هرم ثنائي – Arabic" lang="ar" hreflang="ar" data-title="هرم ثنائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Bipir%C3%A1mide" title="Bipirámide – Asturian" lang="ast" hreflang="ast" data-title="Bipirámide" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%B8%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8%D0%B4%D0%B0" title="Бипирамида – Chuvash" lang="cv" hreflang="cv" data-title="Бипирамида" 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/Dvojjehlan" title="Dvojjehlan – Czech" lang="cs" hreflang="cs" data-title="Dvojjehlan" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Doppelpyramide" title="Doppelpyramide – German" lang="de" hreflang="de" data-title="Doppelpyramide" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Bipir%C3%A1mide" title="Bipirámide – Spanish" lang="es" hreflang="es" data-title="Bipirámide" 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/Dupiramido" title="Dupiramido – Esperanto" lang="eo" hreflang="eo" data-title="Dupiramido" 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/Bipiramide" title="Bipiramide – Basque" lang="eu" hreflang="eu" data-title="Bipiramide" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%88%D9%87%D8%B1%D9%85" title="دوهرم – Persian" lang="fa" hreflang="fa" data-title="دوهرم" 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/Bipyramide" title="Bipyramide – French" lang="fr" hreflang="fr" data-title="Bipyramide" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EA%B0%81%EB%BF%94" title="쌍각뿔 – Korean" lang="ko" hreflang="ko" data-title="쌍각뿔" 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%B5%D6%80%D5%AF%D5%A2%D5%B8%D6%82%D6%80%D5%A3" title="Երկբուրգ – Armenian" lang="hy" hreflang="hy" data-title="Երկբուրգ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Bipiramide" title="Bipiramide – Italian" lang="it" hreflang="it" data-title="Bipiramide" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bipiramide" title="Bipiramide – Dutch" lang="nl" hreflang="nl" data-title="Bipiramide" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8C%E8%A7%92%E9%8C%90" title="双角錐 – Japanese" lang="ja" hreflang="ja" data-title="双角錐" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Dipiramida" title="Dipiramida – Uzbek" lang="uz" hreflang="uz" data-title="Dipiramida" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Bipir%C3%A2mide" title="Bipirâmide – Portuguese" lang="pt" hreflang="pt" data-title="Bipirâmide" 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/Bipiramid%C4%83" title="Bipiramidă – Romanian" lang="ro" hreflang="ro" data-title="Bipiramidă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8%D0%B4%D0%B0" title="Бипирамида – Russian" lang="ru" hreflang="ru" data-title="Бипирамида" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bipiramida" title="Bipiramida – Slovenian" lang="sl" hreflang="sl" data-title="Bipiramida" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B8%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8%D0%B4%D0%B0" title="Бипирамида – Serbian" lang="sr" hreflang="sr" data-title="Бипирамида" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bipyramidi" title="Bipyramidi – Finnish" lang="fi" hreflang="fi" data-title="Bipyramidi" 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/Bipyramid" title="Bipyramid – Swedish" lang="sv" hreflang="sv" data-title="Bipyramid" 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%87%E0%AE%B0%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%88%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%88%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%AE%E0%AF%8D%E0%AE%AA%E0%AF%81" title="இரட்டைப் பட்டைக்கூம்பு – Tamil" lang="ta" hreflang="ta" data-title="இரட்டைப் பட்டைக்கூம்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B5%E0%B8%A3%E0%B8%B0%E0%B8%A1%E0%B8%B4%E0%B8%94%E0%B8%84%E0%B8%B9%E0%B9%88" title="พีระมิดคู่ – Thai" lang="th" hreflang="th" data-title="พีระมิดคู่" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%96%D0%BF%D1%96%D1%80%D0%B0%D0%BC%D1%96%D0%B4%D0%B0" title="Біпіраміда – Ukrainian" lang="uk" hreflang="uk" data-title="Біпіраміда" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%99%E9%8C%90%E9%AB%94" title="雙錐體 – Chinese" lang="zh" hreflang="zh" data-title="雙錐體" data-language-autonym="中文" data-language-local-name="Chinese" 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Polyhedron formed by joining mirroring pyramids base-to-base</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">"Dipyramid" redirects here. For the mountain, see <a href="/wiki/Dipyramid_(Alaska)" title="Dipyramid (Alaska)">Dipyramid (Alaska)</a>.</div> <p class="mw-empty-elt"> </p><p>In geometry, a <b>bipyramid</b>, <b>dipyramid</b>, or <b>double pyramid</b> is a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> formed by fusing two <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a> together <a href="/wiki/Base_(geometry)" title="Base (geometry)">base</a>-to-base. The <a href="/wiki/Polygon" title="Polygon">polygonal</a> base of each pyramid must therefore be the same, and unless otherwise specified the base <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> are usually <a href="/wiki/Coplanar" class="mw-redirect" title="Coplanar">coplanar</a> and a bipyramid is usually <i>symmetric</i>, meaning the two pyramids are <a href="/wiki/Mirror_image" title="Mirror image">mirror images</a> across their common base plane. When each <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a> (<abbr title="plural form">pl.</abbr> apices, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is a <i>right</i> bipyramid;<sup id="cite_ref-right_pyramids_1-0" class="reference"><a href="#cite_note-right_pyramids-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> otherwise it is <i>oblique</i>. When the base is a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>, the bipyramid is also called <i>regular</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_properties">Definition and properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=1" title="Edit section: Definition and properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:492px;max-width:492px"><div class="trow"><div class="tsingle" style="width:216px;max-width:216px"><div class="thumbimage" style="height:174px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Triangular_bipyramid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Triangular_bipyramid.png/214px-Triangular_bipyramid.png" decoding="async" width="214" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Triangular_bipyramid.png/321px-Triangular_bipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Triangular_bipyramid.png/428px-Triangular_bipyramid.png 2x" data-file-width="1028" data-file-height="836" /></a></span></div></div><div class="tsingle" style="width:128px;max-width:128px"><div class="thumbimage" style="height:174px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Square_bipyramid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Square_bipyramid.png/126px-Square_bipyramid.png" decoding="async" width="126" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Square_bipyramid.png/189px-Square_bipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Square_bipyramid.png/252px-Square_bipyramid.png 2x" data-file-width="867" data-file-height="1200" /></a></span></div></div><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage" style="height:174px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Pentagonale_bipiramide.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Pentagonale_bipiramide.png/140px-Pentagonale_bipiramide.png" decoding="async" width="140" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Pentagonale_bipiramide.png/210px-Pentagonale_bipiramide.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Pentagonale_bipiramide.png/280px-Pentagonale_bipiramide.png 2x" data-file-width="717" data-file-height="893" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The <a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">triangular bipyramid</a>, <a href="/wiki/Square_bipyramid" class="mw-redirect" title="Square bipyramid">square bipyramid</a>, and <a href="/wiki/Pentagonal_bipyramid" title="Pentagonal bipyramid">pentagonal bipyramid</a>.</div></div></div></div> <p>A bipyramid is a polyhedron constructed by fusing two <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a> which share the same <a href="/wiki/Polygon" title="Polygon">polygonal</a> <a href="/wiki/Base_(geometry)" title="Base (geometry)">base</a>;<sup id="cite_ref-aarts_2-0" class="reference"><a href="#cite_note-aarts-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> a pyramid is in turn constructed by connecting each vertex of its base to a single new <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> (the <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a>) not lying in the plane of the base, for an <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>gonal base forming <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> triangular faces in addition to the base face. An <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>gonal bipyramid thus has <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 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}"></span> faces, <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 3n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702e054176930a46bb558f22adad5d81f9f0cafd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 3n}"></span> edges, 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 n+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf4207e44c00a9312fe3e8710efbefcd6eafdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+2}"></span> vertices. <span class="anchor" id="Right_and_oblique_bipyramid"></span>More generally, a right pyramid is a pyramid where the apices are on the perpendicular line through the <a href="/wiki/Centroid" title="Centroid">centroid</a> of an arbitrary polygon or the <a href="/wiki/Incenter" title="Incenter">incenter</a> of a <a href="/wiki/Tangential_polygon" title="Tangential polygon">tangential polygon</a>, depending on the source.<sup id="cite_ref-right_pyramids_1-1" class="reference"><a href="#cite_note-right_pyramids-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Likewise, a <i>right bipyramid</i> is a polyhedron constructed by attaching two symmetrical right bipyramid bases; bipyramids whose apices are not on this line are called <i>oblique bipyramids</i>.<sup id="cite_ref-polya_3-0" class="reference"><a href="#cite_note-polya-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>When the two pyramids are mirror images, the bipyramid is called <i>symmetric</i>. It is called <i>regular</i> if its base is a <a href="/wiki/Regular_polygon" title="Regular polygon">regular polygon</a>.<sup id="cite_ref-aarts_2-1" class="reference"><a href="#cite_note-aarts-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> When the base is a regular polygon and the apices are on the perpendicular line through its center (a <i>regular right bipyramid</i>) then all of its faces are <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangles</a>; sometimes the name <i>bipyramid</i> refers specifically to symmetric regular right bipyramids,<sup id="cite_ref-montroll_4-0" class="reference"><a href="#cite_note-montroll-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Examples of such bipyramids are the <a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">triangular bipyramid</a>, <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> (square bipyramid) and <a href="/wiki/Pentagonal_bipyramid" title="Pentagonal bipyramid">pentagonal bipyramid</a>. If all their edges are equal in length, these shapes consist of <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a> faces, making them <a href="/wiki/Deltahedron" title="Deltahedron">deltahedra</a>;<sup id="cite_ref-trigg_5-0" class="reference"><a href="#cite_note-trigg-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-uehara_6-0" class="reference"><a href="#cite_note-uehara-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the triangular bipyramid and the pentagonal bipyramid are <a href="/wiki/Johnson_solid" title="Johnson solid">Johnson solids</a>, and the regular octahedron is a <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solid</a>.<sup id="cite_ref-cromwell_7-0" class="reference"><a href="#cite_note-cromwell-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <figure 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 octahedron is dual to the cube</figcaption></figure> <p>The symmetric regular right bipyramids have <a href="/wiki/Prismatic_symmetry" class="mw-redirect" title="Prismatic symmetry">prismatic symmetry</a>, with <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral symmetry group</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 D_{nh}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{nh}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4493425a37fac6195a7d331d0ac44c901b5461a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.09ex; height:2.509ex;" alt="{\displaystyle D_{nh}}"></span> of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d3d982c0a63d59f04a9ea9aecec75fb107f6a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 4n}"></span>: they are unchanged when rotated <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/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"></span> of a turn around the <a href="/wiki/Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">axis of symmetry</a>, reflected across any plane passing through both apices and a base vertex or both apices and the center of a base edge, or reflected across the mirror plane.<sup id="cite_ref-fsz_8-0" class="reference"><a href="#cite_note-fsz-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Because their faces are transitive under these symmetry transformations, they are <a href="/wiki/Isohedral_figure" title="Isohedral figure">isohedral</a>.<sup id="cite_ref-cpsb_9-0" class="reference"><a href="#cite_note-cpsb-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mclean_10-0" class="reference"><a href="#cite_note-mclean-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> They are the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedra</a> of <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a> and the prisms are the dual of bipyramids as well; the bipyramids vertices correspond to the faces of the prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other, and vice versa.<sup id="cite_ref-sibley_11-0" class="reference"><a href="#cite_note-sibley-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The prisms share the same symmetry as the bipyramids.<sup id="cite_ref-king_12-0" class="reference"><a href="#cite_note-king-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a> is more symmetric still, as its base vertices and apices are indistinguishable and can be exchanged by reflections or <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>; the regular octahedron and its dual, the <a href="/wiki/Cube" title="Cube">cube</a>, have <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a>.<sup id="cite_ref-armstrong_13-0" class="reference"><a href="#cite_note-armstrong-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Volume" title="Volume">volume</a> of a symmetric bipyramid is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{3}}Bh,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>B</mi> <mi>h</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}Bh,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddd0f23d118f5831c84c8e1a146690d24cfea6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.748ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}Bh,}"></span> where <span class="texhtml mvar" style="font-style:italic;">B</span> is the area of the base and <span class="texhtml mvar" style="font-style:italic;">h</span> the height from the base plane to any apex. In the case of a regular <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>sided polygon with side 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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> and whose altitude 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, the volume of such bipyramid is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>6</mn> </mfrac> </mrow> <mi>h</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a74af6ad94d464a55008c58b4d6f1fb533a320c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.466ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Related_and_other_types_of_bipyramid">Related and other types of bipyramid</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=2" title="Edit section: Related and other types of bipyramid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:342px;max-width:342px"><div class="trow"><div class="tsingle" style="width:171px;max-width:171px"><div class="thumbimage" style="height:219px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Concave_quadrilateral_bipyramid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Concave_quadrilateral_bipyramid.png/169px-Concave_quadrilateral_bipyramid.png" decoding="async" width="169" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Concave_quadrilateral_bipyramid.png/254px-Concave_quadrilateral_bipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Concave_quadrilateral_bipyramid.png/338px-Concave_quadrilateral_bipyramid.png 2x" data-file-width="456" data-file-height="592" /></a></span></div><div class="thumbcaption">A concave tetragonal bipyramid</div></div><div class="tsingle" style="width:167px;max-width:167px"><div class="thumbimage" style="height:219px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Asymmetric_hexagonal_bipyramid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Asymmetric_hexagonal_bipyramid.png/165px-Asymmetric_hexagonal_bipyramid.png" decoding="async" width="165" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Asymmetric_hexagonal_bipyramid.png/248px-Asymmetric_hexagonal_bipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Asymmetric_hexagonal_bipyramid.png/330px-Asymmetric_hexagonal_bipyramid.png 2x" data-file-width="1137" data-file-height="1507" /></a></span></div><div class="thumbcaption">An asymmetric hexagonal bipyramid</div></div></div></div></div> <div class="mw-heading mw-heading3"><h3 id="Concave_bipyramids">Concave bipyramids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=3" title="Edit section: Concave bipyramids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>concave bipyramid</i> has a <a href="/wiki/Concave_polygon" title="Concave polygon">concave polygon</a> base, and one example is a concave tetragonal bipyramid or an irregular concave octahedron. A bipyramid with an arbitrary polygonal base could be considered a <i>right</i> bipyramid if the apices are on a line perpendicular to the base passing through the base's <a href="/wiki/Centroid" title="Centroid">centroid</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Asymmetric_bipyramids">Asymmetric bipyramids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=4" title="Edit section: Asymmetric bipyramids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <i>asymmetric bipyramid</i> has apices which are not mirrored across the base plane; for a right bipyramid this only happens if each apex is a different distance from the base. </p><p>The <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> of an asymmetric right <span class="texhtml mvar" style="font-style:italic;">n</span>-gonal bipyramid is an <span class="texhtml mvar" style="font-style:italic;">n</span>-gonal <a href="/wiki/Frustum" title="Frustum">frustum</a>. </p><p>A regular asymmetric right <span class="texhtml mvar" style="font-style:italic;">n</span>-gonal bipyramid has symmetry group <span class="texhtml">C<sub><i>n</i>v</sub></span>, of order <span class="texhtml">2<i>n</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Scalene_triangle_bipyramids">Scalene triangle bipyramids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=5" title="Edit section: Scalene triangle bipyramids"><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:EB1911_Crystallography_Fig._46_Ditetragonal_Bipyramid.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/fb/EB1911_Crystallography_Fig._46_Ditetragonal_Bipyramid.jpg" decoding="async" width="149" height="267" class="mw-file-element" data-file-width="149" data-file-height="267" /></a><figcaption>Example: ditetragonal bipyramid (<span class="texhtml">2<i>n</i> = 2×4</span>)</figcaption></figure> <p>An isotoxal right (symmetric) <b>di-<span class="texhtml mvar" style="font-style:italic;">n</span>-gonal bipyramid</b> is a right (symmetric) <span class="texhtml"><b>2</b><i>n</i></span>-gonal bipyramid with an <a href="/wiki/Isotoxal_figure" title="Isotoxal figure"><i>isotoxal</i></a> flat polygon base: its <span class="texhtml">2<i>n</i></span> basal vertices are coplanar, but alternate in two <a href="/wiki/Radius" title="Radius">radii</a>. </p><p>All its faces are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> <a href="/wiki/Scalene_triangle" class="mw-redirect" title="Scalene triangle">scalene triangles</a>, and it is <a href="/wiki/Isohedral_figure" title="Isohedral figure">isohedral</a>. It can be seen as another type of a right symmetric di-<span class="texhtml mvar" style="font-style:italic;">n</span>-gonal <a href="#Scalenohedra"><i>scalenohedron</i></a>, with an isotoxal flat polygon base. </p><p>An isotoxal right (symmetric) di-<span class="texhtml mvar" style="font-style:italic;">n</span>-gonal bipyramid has <span class="texhtml mvar" style="font-style:italic;">n</span> two-fold rotation axes through opposite basal vertices, <span class="texhtml mvar" style="font-style:italic;">n</span> reflection planes through opposite apical edges, an <span class="texhtml mvar" style="font-style:italic;">n</span>-fold rotation axis through apices, a reflection plane through base, and an <span class="texhtml mvar" style="font-style:italic;">n</span>-fold <a href="/wiki/Improper_rotation" title="Improper rotation">rotation-reflection</a> axis through apices,<sup id="cite_ref-tulane_14-0" class="reference"><a href="#cite_note-tulane-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> representing symmetry group <span class="texhtml">D<sub><i>n</i>h</sub>, [<i>n</i>,2], (*22<i>n</i>),</span> of order <span class="texhtml">4<i>n</i></span>. (The reflection about the base plane corresponds to the <span class="texhtml">0°</span> rotation-reflection. If <span class="texhtml mvar" style="font-style:italic;">n</span> is even, then there is an <a href="/wiki/Inversion_symmetry" class="mw-redirect" title="Inversion symmetry">inversion symmetry</a> about the center, corresponding to the <span class="texhtml">180°</span> rotation-reflection.) </p><p>Example with <span class="texhtml">2<i>n</i> = 2×3</span>: </p> <dl><dd>An isotoxal right (symmetric) ditrigonal bipyramid has three similar vertical planes of symmetry, intersecting in a (vertical) <span class="texhtml">3</span>-fold rotation axis; perpendicular to them is a fourth plane of symmetry (horizontal); at the intersection of the three vertical planes with the horizontal plane are three similar (horizontal) <span class="texhtml">2</span>-fold rotation axes; there is no center of inversion symmetry,<sup id="cite_ref-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_ditrigonal_bipyramidal_class,_p._581_(p._603_on_Wikisource)_15-0" class="reference"><a href="#cite_note-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_ditrigonal_bipyramidal_class,_p._581_(p._603_on_Wikisource)-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> but there is a <a href="/wiki/Center_of_symmetry" class="mw-redirect" title="Center of symmetry">center of symmetry</a>: the intersection point of the four axes.</dd></dl> <p>Example with <span class="texhtml">2<i>n</i> = 2×4</span>: </p> <dl><dd>An isotoxal right (symmetric) ditetragonal bipyramid has four vertical planes of symmetry of two kinds, intersecting in a (vertical) <span class="texhtml">4</span>-fold rotation axis; perpendicular to them is a fifth plane of symmetry (horizontal); at the intersection of the four vertical planes with the horizontal plane are four (horizontal) <span class="texhtml">2</span>-fold rotation axes of two kinds, each perpendicular to a plane of symmetry; two vertical planes bisect the angles between two horizontal axes; and there is a centre of inversion symmetry.<sup id="cite_ref-FOOTNOTESpencer19112._Tegragonal_system,_holosymmetric_class,_fig._46,_p._577_(p._599_on_Wikisource)_16-0" class="reference"><a href="#cite_note-FOOTNOTESpencer19112._Tegragonal_system,_holosymmetric_class,_fig._46,_p._577_(p._599_on_Wikisource)-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Double example: </p> <ul><li>The bipyramid with isotoxal <span class="texhtml">2×2</span>-gon base vertices <span class="texhtml mvar" style="font-style:italic;">U, U', V, V'</span> and right symmetric apices <span class="texhtml mvar" style="font-style:italic;">A, A'</span><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{alignedat}{5}U&=(1,0,0),&\quad V&=(0,2,0),&\quad A&=(0,0,1),\\U'&=(-1,0,0),&\quad V'&=(0,-2,0),&\quad A'&=(0,0,-1),\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>V</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}U&=(1,0,0),&\quad V&=(0,2,0),&\quad A&=(0,0,1),\\U'&=(-1,0,0),&\quad V'&=(0,-2,0),&\quad A'&=(0,0,-1),\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140a029cc4f0b255f1e9c448df76bf18d401649c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.706ex; height:6.176ex;" alt="{\displaystyle {\begin{alignedat}{5}U&=(1,0,0),&\quad V&=(0,2,0),&\quad A&=(0,0,1),\\U'&=(-1,0,0),&\quad V'&=(0,-2,0),&\quad A'&=(0,0,-1),\end{alignedat}}}"></span> has its faces isosceles. Indeed: <ul><li>Upper apical edge lengths:<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{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {2}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {5}}\,;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {2}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {5}}\,;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ed54ba87f7378e143c3f37b5bc398c4d92b45b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.467ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {2}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {5}}\,;\end{aligned}}}"></span></li> <li>Base edge lengths: <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 {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>U</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af86217b29aca302aa8fb771b9c0815612858a29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.789ex; height:3.509ex;" alt="{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;}"></span></li> <li>Lower apical edge lengths (equal to upper edge lengths):<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{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {2}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {5}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {2}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {5}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71238acf7697d5dc9481fcb4b8668db0b43f6096" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.836ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {2}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {5}}\,.\end{aligned}}}"></span></li></ul></li></ul> <ul><li>The bipyramid with same base vertices, but with right symmetric apices <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{aligned}A&=(0,0,2),\\A'&=(0,0,-2),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=(0,0,2),\\A'&=(0,0,-2),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3cdf64a35989f5733beeb545cf05a5ab7f999b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.097ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}A&=(0,0,2),\\A'&=(0,0,-2),\end{aligned}}}"></span> also has its faces isosceles. Indeed: <ul><li>Upper apical edge lengths:<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{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {2}}\,;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {2}}\,;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39214c8e5884badd6bd6f588b3dc8f7e63ab97e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.629ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {2}}\,;\end{aligned}}}"></span></li> <li>Base edge length (equal to previous example): <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 {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>U</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af86217b29aca302aa8fb771b9c0815612858a29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.789ex; height:3.509ex;" alt="{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;}"></span></li> <li>Lower apical edge lengths (equal to upper edge lengths):<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{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {5}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {5}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d42f15bc7685f8f26d0b5a7bf18fc0bc3d0b92e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.998ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {5}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}}"></span></li></ul></li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg/220px-EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg/330px-EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg/440px-EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg 2x" data-file-width="498" data-file-height="256" /></a><figcaption>Examples of rhombic bipyramids</figcaption></figure> <p>In <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>, isotoxal right (symmetric) didigonal<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.<sup id="cite_ref-tulane_14-2" class="reference"><a href="#cite_note-tulane-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-uwgb_17-1" class="reference"><a href="#cite_note-uwgb-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Scalenohedra">Scalenohedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=6" title="Edit section: Scalenohedra"><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:EB1911_Crystallography_Fig._68.%E2%80%94Scalenohedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/20/EB1911_Crystallography_Fig._68.%E2%80%94Scalenohedron.jpg" decoding="async" width="123" height="278" class="mw-file-element" data-file-width="123" data-file-height="278" /></a><figcaption>Example: ditrigonal scalenohedron (<span class="texhtml">2<i>n</i> = 2×3</span>)</figcaption></figure> <p>A <b>scalenohedron</b> is similar to a bipyramid; the difference is that the scalenohedra has a zig-zag pattern in the middle edges.<sup id="cite_ref-kp_19-0" class="reference"><a href="#cite_note-kp-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>It has two apices and <span class="texhtml">2<i>n</i></span> basal vertices, <span class="texhtml">4<i>n</i></span> faces, and <span class="texhtml">6<i>n</i></span> edges; it is topologically identical to a <span class="texhtml">2<i>n</i></span>-gonal bipyramid, but its <span class="texhtml">2<i>n</i></span> basal vertices alternate in two rings above and below the center.<sup id="cite_ref-uwgb_17-2" class="reference"><a href="#cite_note-uwgb-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>All its faces are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> <a href="/wiki/Scalene_triangle" class="mw-redirect" title="Scalene triangle">scalene triangles</a>, and it is <a href="/wiki/Isohedral_figure" title="Isohedral figure">isohedral</a>. It can be seen as another type of a right symmetric di-<span class="texhtml mvar" style="font-style:italic;">n</span>-gonal bipyramid, with a regular zigzag skew polygon base. </p><p>A regular right symmetric di-<span class="texhtml mvar" style="font-style:italic;">n</span>-gonal scalenohedron has <span class="texhtml mvar" style="font-style:italic;">n</span> two-fold rotation axes through opposite basal mid-edges, <span class="texhtml mvar" style="font-style:italic;">n</span> reflection planes through opposite apical edges, an <span class="texhtml mvar" style="font-style:italic;">n</span>-fold rotation axis through apices, and a <span class="texhtml"><b>2</b><i>n</i></span>-fold <a href="/wiki/Improper_rotation" title="Improper rotation">rotation-reflection</a> axis through apices (about which <span class="texhtml"><b>1</b><i>n</i></span> rotations-reflections globally preserve the solid),<sup id="cite_ref-tulane_14-3" class="reference"><a href="#cite_note-tulane-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> representing symmetry group <span class="texhtml">D<sub><i>n</i>v</sub> = D<sub><i>n</i>d</sub>, [2<sup>+</sup>,2<i>n</i>], (2*<i>n</i>),</span> of order <span class="texhtml">4<i>n</i></span>. (If <span class="texhtml mvar" style="font-style:italic;">n</span> is odd, then there is an <a href="/wiki/Inversion_symmetry" class="mw-redirect" title="Inversion symmetry">inversion symmetry</a> about the center, corresponding to the <span class="texhtml">180°</span> rotation-reflection.) </p><p>Example with <span class="texhtml">2<i>n</i> = 2×3</span>: </p> <dl><dd>A regular right symmetric ditrigonal scalenohedron has three similar vertical planes of symmetry inclined to one another at <span class="texhtml">60°</span> and intersecting in a (vertical) <span class="texhtml">3</span>-fold rotation axis, three similar horizontal <span class="texhtml">2</span>-fold rotation axes, each perpendicular to a plane of symmetry, a center of inversion symmetry,<sup id="cite_ref-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_holosymmetric_class,_fig._68,_p._580_(p._602_on_Wikisource)_20-0" class="reference"><a href="#cite_note-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_holosymmetric_class,_fig._68,_p._580_(p._602_on_Wikisource)-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> and a vertical <span class="texhtml"><b>6</b></span>-fold rotation-reflection axis.</dd></dl> <p>Example with <span class="texhtml">2<i>n</i> = 2×2</span>: </p> <dl><dd>A regular right symmetric didigonal scalenohedron has only one vertical and two horizontal <span class="texhtml">2</span>-fold rotation axes, two vertical planes of symmetry, which bisect the angles between the horizontal pair of axes, and a vertical <span class="texhtml"><b>4</b></span>-fold rotation-reflection axis;<sup id="cite_ref-FOOTNOTESpencer19112._Tetragonal_system,_scalenohedral_class,_fig._51,_p._577_(p._599_on_Wikisource)_21-0" class="reference"><a href="#cite_note-FOOTNOTESpencer19112._Tetragonal_system,_scalenohedral_class,_fig._51,_p._577_(p._599_on_Wikisource)-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> it has no center of inversion symmetry.</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:EB1911_Crystallography_Figs._50_%26_51.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/EB1911_Crystallography_Figs._50_%26_51.jpg/220px-EB1911_Crystallography_Figs._50_%26_51.jpg" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/EB1911_Crystallography_Figs._50_%26_51.jpg/330px-EB1911_Crystallography_Figs._50_%26_51.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/EB1911_Crystallography_Figs._50_%26_51.jpg/440px-EB1911_Crystallography_Figs._50_%26_51.jpg 2x" data-file-width="465" data-file-height="239" /></a><figcaption>Examples of disphenoids and of an <span class="texhtml">8</span>-faced scalenohedron</figcaption></figure> <p>For at most two particular values 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 z_{A}=|z_{A'}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{A}=|z_{A'}|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8ac843b16a414755d271ba0503c81db74ce14d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.663ex; height:2.843ex;" alt="{\displaystyle z_{A}=|z_{A'}|,}"></span> the faces of such a <b>scaleno</b>hedron may be <a href="/wiki/Isosceles_triangle" title="Isosceles triangle"><b>isosceles</b></a>. </p><p>Double example: </p> <ul><li>The scalenohedron with regular zigzag skew <span class="texhtml">2×2</span>-gon base vertices <span class="texhtml mvar" style="font-style:italic;">U, U', V, V'</span> and right symmetric apices <span class="texhtml mvar" style="font-style:italic;">A, A'</span><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{alignedat}{5}U&=(3,0,2),&\quad V&=(0,3,-2),&\quad A&=(0,0,3),\\U'&=(-3,0,2),&\quad V'&=(0,-3,-2),&\quad A'&=(0,0,-3),\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>V</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}U&=(3,0,2),&\quad V&=(0,3,-2),&\quad A&=(0,0,3),\\U'&=(-3,0,2),&\quad V'&=(0,-3,-2),&\quad A'&=(0,0,-3),\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29418714fb10ffdc701c6c4effe853f43d78c46" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.514ex; height:6.176ex;" alt="{\displaystyle {\begin{alignedat}{5}U&=(3,0,2),&\quad V&=(0,3,-2),&\quad A&=(0,0,3),\\U'&=(-3,0,2),&\quad V'&=(0,-3,-2),&\quad A'&=(0,0,-3),\end{alignedat}}}"></span> has its faces isosceles. Indeed: <ul><li>Upper apical edge lengths:<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{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {10}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {34}}\,;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {10}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {34}}\,;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d07abca564b334ebf5464dc9a141e045ff1397" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.629ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {10}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {34}}\,;\end{aligned}}}"></span></li> <li>Base edge length:<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 {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>U</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2fa1274484ab5c45d2cf603d42b749cdf2b326" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.951ex; height:3.509ex;" alt="{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;}"></span></li> <li>Lower apical edge lengths (equal to upper edge lengths swapped):<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{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {34}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {10}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {34}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {10}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61d1de369b0a90747a09578d1d1d03cb7161235" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.998ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {34}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {10}}\,.\end{aligned}}}"></span></li></ul></li></ul> <ul><li>The scalenohedron with same base vertices, but with right symmetric apices<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{aligned}A&=(0,0,7),\\A'&=(0,0,-7),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=(0,0,7),\\A'&=(0,0,-7),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb4d2448cb2e7da37fe341edd9885c01693fea3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.097ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}A&=(0,0,7),\\A'&=(0,0,-7),\end{aligned}}}"></span> also has its faces isosceles. Indeed: <ul><li>Upper apical edge lengths:<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{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=3{\sqrt {10}}\,;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=3{\sqrt {10}}\,;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06ffa51557cb5132790f07fd96756e6595c406ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.791ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=3{\sqrt {10}}\,;\end{aligned}}}"></span></li> <li>Base edge length (equal to previous example): <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 {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>U</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2fa1274484ab5c45d2cf603d42b749cdf2b326" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.951ex; height:3.509ex;" alt="{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;}"></span></li> <li>Lower apical edge lengths (equal to upper edge lengths swapped):<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{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {34}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {34}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e8f672aac8e87848cd9f096966fc3c1634b13e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.085ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {34}}\,.\end{aligned}}}"></span></li></ul></li></ul> <p>In <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>, regular right symmetric didigonal (<span class="texhtml">8</span>-faced) and ditrigonal (<span class="texhtml">12</span>-faced) scalenohedra exist.<sup id="cite_ref-tulane_14-4" class="reference"><a href="#cite_note-tulane-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-uwgb_17-3" class="reference"><a href="#cite_note-uwgb-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The smallest geometric scalenohedra have eight faces, and are topologically identical to the regular <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>. In this case (<span class="texhtml">2<i>n</i> = 2×2</span>), in crystallography, a regular right symmetric didigonal (<span class="texhtml">8</span>-faced) scalenohedron is called a <i>tetragonal scalenohedron</i>.<sup id="cite_ref-tulane_14-5" class="reference"><a href="#cite_note-tulane-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-uwgb_17-4" class="reference"><a href="#cite_note-uwgb-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Let us temporarily focus on the regular right symmetric <span class="texhtml">8</span>-faced scalenohedra with <span class="texhtml"><i>h</i> = <i>r</i>,</span> i.e. <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 z_{A}=|z_{A'}|=x_{U}=|x_{U'}|=y_{V}=|y_{V'}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{A}=|z_{A'}|=x_{U}=|x_{U'}|=y_{V}=|y_{V'}|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9866c882a8fc1d9431cda43fc90bb9a093d0ed37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.755ex; height:2.843ex;" alt="{\displaystyle z_{A}=|z_{A'}|=x_{U}=|x_{U'}|=y_{V}=|y_{V'}|.}"></span> Their two apices can be represented as <span class="texhtml mvar" style="font-style:italic;">A, A'</span> and their four basal vertices as <span class="texhtml mvar" style="font-style:italic;">U, U', V, V'</span>: <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{alignedat}{5}U&=(1,0,z),&\quad V&=(0,1,-z),&\quad A&=(0,0,1),\\U'&=(-1,0,z),&\quad V'&=(0,-1,-z),&\quad A'&=(0,0,-1),\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>V</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}U&=(1,0,z),&\quad V&=(0,1,-z),&\quad A&=(0,0,1),\\U'&=(-1,0,z),&\quad V'&=(0,-1,-z),&\quad A'&=(0,0,-1),\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f370105171e1b19a4851a82559fd2bb0c1d02306" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.366ex; height:6.176ex;" alt="{\displaystyle {\begin{alignedat}{5}U&=(1,0,z),&\quad V&=(0,1,-z),&\quad A&=(0,0,1),\\U'&=(-1,0,z),&\quad V'&=(0,-1,-z),&\quad A'&=(0,0,-1),\end{alignedat}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">z</span> is a parameter between <span class="texhtml">0</span> and <span class="texhtml">1</span>. </p><p>At <span class="texhtml"><i>z</i> = 0</span>, it is a regular octahedron; at <span class="texhtml"><i>z</i> = 1</span>, it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it a <a href="/wiki/Disphenoid" title="Disphenoid">disphenoid</a>; for <span class="texhtml"><i>z</i> > 1</span>, it is concave. </p> <table class="wikitable"> <caption style="text-align:center;">Example: geometric variations with regular right symmetric 8-faced scalenohedra: </caption> <tbody><tr> <th><span class="texhtml"><i>z</i> = 0.1</span> </th> <th><span class="texhtml"><i>z</i> = 0.25</span> </th> <th><span class="texhtml"><i>z</i> = 0.5</span> </th> <th><span class="texhtml"><i>z</i> = 0.95</span> </th> <th><span class="texhtml"><i>z</i> = 1.5</span> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:4-scalenohedron-01.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/4-scalenohedron-01.png/120px-4-scalenohedron-01.png" decoding="async" width="120" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/4-scalenohedron-01.png/180px-4-scalenohedron-01.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/4-scalenohedron-01.png/240px-4-scalenohedron-01.png 2x" data-file-width="1597" data-file-height="1991" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:4-scalenohedron-025.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-scalenohedron-025.png/120px-4-scalenohedron-025.png" decoding="async" width="120" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-scalenohedron-025.png/180px-4-scalenohedron-025.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-scalenohedron-025.png/240px-4-scalenohedron-025.png 2x" data-file-width="1557" data-file-height="1983" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:4-scalenohedron-05.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/4-scalenohedron-05.png/120px-4-scalenohedron-05.png" decoding="async" width="120" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/4-scalenohedron-05.png/180px-4-scalenohedron-05.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/4-scalenohedron-05.png/240px-4-scalenohedron-05.png 2x" data-file-width="1499" data-file-height="1840" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:4-scalenohedron-095.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/4-scalenohedron-095.png/120px-4-scalenohedron-095.png" decoding="async" width="120" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/4-scalenohedron-095.png/180px-4-scalenohedron-095.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/4-scalenohedron-095.png/240px-4-scalenohedron-095.png 2x" data-file-width="1427" data-file-height="1624" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:4-scalenohedron-15.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/4-scalenohedron-15.png/120px-4-scalenohedron-15.png" decoding="async" width="120" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/4-scalenohedron-15.png/180px-4-scalenohedron-15.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/4-scalenohedron-15.png/240px-4-scalenohedron-15.png 2x" data-file-width="1390" data-file-height="1787" /></a></span> </td></tr></tbody></table> <p>If the <span class="texhtml">2<i>n</i></span>-gon base is both <a href="/wiki/Isotoxal_figure" title="Isotoxal figure">isotoxal</a> in-out and <a href="/wiki/Skew_polygon" title="Skew polygon">zigzag skew</a>, then <b>not</b> all faces of the isotoxal right symmetric scalenohedron are congruent. </p><p>Example with five different edge lengths: </p> <ul><li>The scalenohedron with isotoxal in-out zigzag skew <span class="texhtml">2×2</span>-gon base vertices <span class="texhtml mvar" style="font-style:italic;">U, U', V, V'</span> and right symmetric apices <span class="texhtml mvar" style="font-style:italic;">A, A'</span> <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{alignedat}{5}U&=(1,0,1),&\quad V&=(0,2,-1),&\quad A&=(0,0,3),\\U'&=(-1,0,1),&\quad V'&=(0,-2,-1),&\quad A'&=(0,0,-3),\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>V</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}U&=(1,0,1),&\quad V&=(0,2,-1),&\quad A&=(0,0,3),\\U'&=(-1,0,1),&\quad V'&=(0,-2,-1),&\quad A'&=(0,0,-3),\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b55bc6618cbe84786266ded2e2c19473051103" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.514ex; height:6.176ex;" alt="{\displaystyle {\begin{alignedat}{5}U&=(1,0,1),&\quad V&=(0,2,-1),&\quad A&=(0,0,3),\\U'&=(-1,0,1),&\quad V'&=(0,-2,-1),&\quad A'&=(0,0,-3),\end{alignedat}}}"></span> has congruent scalene upper faces, and congruent scalene lower faces, but not all its faces are congruent. Indeed: <ul><li>Upper apical edge lengths:<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{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {5}}\,;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {5}}\,;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27d8ec54eca48cf9b6edd5aa212a116703e079f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.629ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {5}}\,;\end{aligned}}}"></span></li> <li>Base edge length:<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 {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>U</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66c43f6d21d2d0300c62f4c9dda5f1194945f6eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.466ex; height:3.509ex;" alt="{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3;}"></span></li> <li>Lower apical edge lengths:<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{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {17}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>17</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {17}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ef723e1c481d0fed73e8e10c58d890a2463c97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.998ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {17}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}}"></span></li></ul></li></ul> <p>For some particular values of <span class="texhtml"><i>z<sub>A</sub></i> = |<i>z<sub>A'</sub></i>|</span>, half the faces of such a <b>scaleno</b>hedron may be <a href="/wiki/Isosceles_triangle" title="Isosceles triangle"><b>isosceles</b></a> or <a href="/wiki/Equilateral_triangle" title="Equilateral triangle"><b>equilateral</b></a>. </p><p>Example with three different edge lengths: </p> <ul><li>The scalenohedron with isotoxal in-out zigzag skew <span class="texhtml">2×2</span>-gon base vertices <span class="texhtml mvar" style="font-style:italic;">U, U', V, V'</span> and right symmetric apices <span class="texhtml mvar" style="font-style:italic;">A, A'</span> <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{alignedat}{5}U&=(3,0,2),&\quad V&=\left(0,{\sqrt {65}},-2\right),&\quad A&=(0,0,7),\\U'&=(-3,0,2),&\quad V'&=\left(0,-{\sqrt {65}},-2\right),&\quad A'&=(0,0,-7),\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>65</mn> </msqrt> </mrow> <mo>,</mo> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>U</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>V</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>65</mn> </msqrt> </mrow> <mo>,</mo> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mspace width="1em" /> <msup> <mi>A</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}U&=(3,0,2),&\quad V&=\left(0,{\sqrt {65}},-2\right),&\quad A&=(0,0,7),\\U'&=(-3,0,2),&\quad V'&=\left(0,-{\sqrt {65}},-2\right),&\quad A'&=(0,0,-7),\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8eeccf24dfda2867faf5f8be26cbe117599d404" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:57.32ex; height:6.843ex;" alt="{\displaystyle {\begin{alignedat}{5}U&=(3,0,2),&\quad V&=\left(0,{\sqrt {65}},-2\right),&\quad A&=(0,0,7),\\U'&=(-3,0,2),&\quad V'&=\left(0,-{\sqrt {65}},-2\right),&\quad A'&=(0,0,-7),\end{alignedat}}}"></span> has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed: <ul><li>Upper apical edge lengths:<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{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {146}}\,;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>34</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>146</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {146}}\,;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf1385e6a71316ef300aa5d34d24f491eaf6e8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.791ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {146}}\,;\end{aligned}}}"></span></li> <li>Base edge length:<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 {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3{\sqrt {10}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>U</mi> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>V</mi> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>U</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>V</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3{\sqrt {10}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71999f35da89016f2b3ae00e253cf5fa33af961" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.113ex; height:3.509ex;" alt="{\displaystyle {\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3{\sqrt {10}}\,;}"></span></li> <li>Lower apical edge length(s): <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{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=3{\sqrt {10}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>U</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>U</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <mi>V</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>V</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=3{\sqrt {10}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27da30e791e739086b427cd33fa737671c6d1228" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.161ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=3{\sqrt {10}}\,.\end{aligned}}}"></span></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Star_bipyramids">Star bipyramids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=7" title="Edit section: Star bipyramids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b><i>star</i> bipyramid</b> has a <a href="/wiki/Star_polygon" title="Star polygon">star polygon</a> base, and is self-intersecting.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>A regular right symmetric star bipyramid has <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles</a> triangle faces, and is <a href="/wiki/Isohedral_figure" title="Isohedral figure">isohedral</a>. </p><p>A <span class="texhtml"><i>p</i>/<i>q</i></span>-bipyramid has <a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagram</a> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8f/CDel_rat.png" decoding="async" width="3" height="23" class="mw-file-element" data-file-width="3" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span>. </p> <table class="wikitable"> <caption style="text-align:center;">Example star bipyramids: </caption> <tbody><tr align="center"> <th>Base </th> <th><a href="/wiki/Pentagrammic_bipyramid" class="mw-redirect" title="Pentagrammic bipyramid">5/2</a>-gon </th> <th>7/2-gon </th> <th>7/3-gon </th> <th>8/3-gon </th></tr> <tr align="center"> <th>Image </th> <td><span typeof="mw:File"><a href="/wiki/File:Pentagram_Dipyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Pentagram_Dipyramid.png/100px-Pentagram_Dipyramid.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Pentagram_Dipyramid.png/150px-Pentagram_Dipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Pentagram_Dipyramid.png/200px-Pentagram_Dipyramid.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:7-2_dipyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/7-2_dipyramid.png/125px-7-2_dipyramid.png" decoding="async" width="125" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/7-2_dipyramid.png/188px-7-2_dipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/7-2_dipyramid.png/250px-7-2_dipyramid.png 2x" data-file-width="1024" data-file-height="802" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:7-3_dipyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/7-3_dipyramid.png/125px-7-3_dipyramid.png" decoding="async" width="125" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/7-3_dipyramid.png/188px-7-3_dipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/7-3_dipyramid.png/250px-7-3_dipyramid.png 2x" data-file-width="1024" data-file-height="802" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:8-3_dipyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/8-3_dipyramid.png/125px-8-3_dipyramid.png" decoding="async" width="125" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/8-3_dipyramid.png/188px-8-3_dipyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/8-3_dipyramid.png/250px-8-3_dipyramid.png 2x" data-file-width="1024" data-file-height="802" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="4-polytopes_with_bipyramidal_cells">4-polytopes with bipyramidal cells</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=8" title="Edit section: 4-polytopes with bipyramidal cells"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Dual_polyhedron#Dual_polytopes_and_tessellations" title="Dual polyhedron">dual</a> of the <a href="/wiki/Rectification_(geometry)" title="Rectification (geometry)">rectification</a> of each <a href="/wiki/Convex_regular_4-polytope" class="mw-redirect" title="Convex regular 4-polytope">convex regular 4-polytopes</a> is a <a href="/wiki/Cell-transitive" class="mw-redirect" title="Cell-transitive">cell-transitive</a> <a href="/wiki/4-polytope" title="4-polytope">4-polytope</a> with bipyramidal cells. In the following: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">A</span> is the apex vertex of the bipyramid;</li> <li><span class="texhtml mvar" style="font-style:italic;">E</span> is an equator vertex;</li> <li><span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">EE</span></span> is the distance between adjacent vertices on the equator (equal to 1);</li> <li><span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">AE</span></span> is the apex-to-equator edge length;</li> <li><span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">AA</span></span> is the distance between the apices.</li></ul> <p>The bipyramid 4-polytope will have <span class="texhtml mvar" style="font-style:italic;">V<sub>A</sub></span> vertices where the apices of <span class="texhtml mvar" style="font-style:italic;">N<sub>A</sub></span> bipyramids meet. It will have <span class="texhtml mvar" style="font-style:italic;">V<sub>E</sub></span> vertices where the type <span class="texhtml mvar" style="font-style:italic;">E</span> vertices of <span class="texhtml mvar" style="font-style:italic;">N<sub>E</sub></span> bipyramids meet. </p> <ul><li><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_{\overline {AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\overline {AE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2cd5cae7b5a4c16dfd51a1cb46a2f5f5b5bb508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.708ex; height:3.009ex;" alt="{\displaystyle N_{\overline {AE}}}"></span>⁠</span> bipyramids meet along each type <span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">AE</span></span> edge.</li> <li><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_{\overline {EE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\overline {EE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2d5e25c9c0a7104497dd5dc278724a1b44f61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.731ex; height:3.009ex;" alt="{\displaystyle N_{\overline {EE}}}"></span>⁠</span> bipyramids meet along each type <span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">EE</span></span> edge.</li> <li><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 C_{\overline {AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\overline {AE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d99cdcfd68a7ed8c4882de813c7306b70f0e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.503ex; height:3.009ex;" alt="{\displaystyle C_{\overline {AE}}}"></span>⁠</span> is the cosine of the <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> along an <span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">AE</span></span> edge.</li> <li><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 C_{\overline {EE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\overline {EE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003bc4735fd984ae69cfc806fcda1daaa78f0bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.526ex; height:3.009ex;" alt="{\displaystyle C_{\overline {EE}}}"></span>⁠</span> is the cosine of the dihedral angle along an <span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">EE</span></span> edge.</li></ul> <p>As cells must fit around an edge, <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{aligned}N_{\overline {EE}}\arccos C_{\overline {EE}}&\leq 2\pi ,\\[4pt]N_{\overline {AE}}\arccos C_{\overline {AE}}&\leq 2\pi .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> <mi>arccos</mi> <mo>⁡</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> <mi>arccos</mi> <mo>⁡</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}N_{\overline {EE}}\arccos C_{\overline {EE}}&\leq 2\pi ,\\[4pt]N_{\overline {AE}}\arccos C_{\overline {AE}}&\leq 2\pi .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542914fb2d64063185d43f3abbf092d22df436a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.24ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}N_{\overline {EE}}\arccos C_{\overline {EE}}&\leq 2\pi ,\\[4pt]N_{\overline {AE}}\arccos C_{\overline {AE}}&\leq 2\pi .\end{aligned}}}"></span> </p> <table class="wikitable"> <caption style="text-align:center;">4-polytopes with bipyramidal cells </caption> <tbody><tr> <th colspan="9">4-polytope properties </th> <th colspan="6">Bipyramid properties </th></tr> <tr align="center"> <th>Dual of <br /> rectified <br /> polytope </th> <th><a href="/wiki/Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter<br />diagram</a> </th> <th>Cells </th> <th><span class="texhtml mvar" style="font-style:italic;">V<sub>A</sub></span> </th> <th><span class="texhtml mvar" style="font-style:italic;">V<sub>E</sub></span> </th> <th><span class="texhtml mvar" style="font-style:italic;">N<sub>A</sub></span> </th> <th><span class="texhtml mvar" style="font-style:italic;">N<sub>E</sub></span> </th> <th style="padding:0.2em;"><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_{\overline {\!AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mspace width="negativethinmathspace" /> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\overline {\!AE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f57012bd949485e03d540b7353d25cfd0f1f9424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.321ex; height:3.009ex;" alt="{\displaystyle N_{\overline {\!AE}}}"></span>⁠</span> </th> <th style="padding:0.2em;"><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_{\overline {\!EE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mspace width="negativethinmathspace" /> <mi>E</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\overline {\!EE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d9afc0b438f92093baabd8cc46125a7c12aeff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.344ex; height:3.009ex;" alt="{\displaystyle N_{\overline {\!EE}}}"></span>⁠</span> </th> <th>Bipyramid <br /> cell </th> <th>Coxeter<br />diagram </th> <th><span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">AA</span></span> </th> <th><span style="text-decoration:overline;"><span class="texhtml mvar" style="font-style:italic;">AE</span></span><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </th> <th><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 C_{\overline {AE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\overline {AE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d99cdcfd68a7ed8c4882de813c7306b70f0e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.503ex; height:3.009ex;" alt="{\displaystyle C_{\overline {AE}}}"></span>⁠</span> </th> <th><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 C_{\overline {EE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>E</mi> <mi>E</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\overline {EE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003bc4735fd984ae69cfc806fcda1daaa78f0bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.526ex; height:3.009ex;" alt="{\displaystyle C_{\overline {EE}}}"></span>⁠</span> </th></tr> <tr align="center"> <td><a href="/wiki/Rectified_5-cell" title="Rectified 5-cell">R. 5-cell</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>10 </td> <td>5 </td> <td>5 </td> <td>4 </td> <td>6 </td> <td>3 </td> <td>3 </td> <td><a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">Triangular</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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 {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0e30d9a795c6a0635f89ec69e4ef7ed13a0d14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {2}{3}}}"></span> </td> <td>0.667 </td> <td><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}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05614140111b41f1fc23c676a0dd348ff95c1dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {1}{7}}}"></span> </td> <td><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}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05614140111b41f1fc23c676a0dd348ff95c1dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {1}{7}}}"></span> </td></tr> <tr align="center"> <td><a href="/wiki/Rectified_tesseract" title="Rectified tesseract">R. tesseract</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>32 </td> <td>16 </td> <td>8 </td> <td>4 </td> <td>12 </td> <td>3 </td> <td>4 </td> <td><a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">Triangular</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\sqrt {2}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbcc8cc8052f7e07670f6d91798b03c0a1153455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.027ex; height:4.343ex;" alt="{\textstyle {\frac {\sqrt {2}}{3}}}"></span> </td> <td>0.624 </td> <td><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 {2}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {2}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cca3eb2d6a90dc0c458eac375d84632715cb414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {2}{5}}}"></span> </td> <td><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}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc2c65769cb58a031b393c3c3c92cf7fec8e787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {1}{5}}}"></span> </td></tr> <tr align="center"> <td><a href="/wiki/Rectified_24-cell" title="Rectified 24-cell">R. 24-cell</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>96 </td> <td>24 </td> <td>24 </td> <td>8 </td> <td>12 </td> <td>4 </td> <td>3 </td> <td><a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">Triangular</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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 {2{\sqrt {2}}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {2{\sqrt {2}}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74d5202dc1dc2faa84fe41e53f28c49e34904b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.849ex; height:4.343ex;" alt="{\textstyle {\frac {2{\sqrt {2}}}{3}}}"></span> </td> <td>0.745 </td> <td><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}{11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{11}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adb90e0f277d7be1cf2803a24e77607ae2833f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.48ex; height:3.509ex;" alt="{\textstyle {\frac {1}{11}}}"></span> </td> <td><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 {5}{11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>11</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {5}{11}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22811d71d626e867d2d172cdfb535beb5b7ace4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.288ex; height:3.509ex;" alt="{\textstyle -{\frac {5}{11}}}"></span> </td></tr> <tr align="center"> <td><a href="/wiki/Rectified_120-cell" title="Rectified 120-cell">R. 120-cell</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>1200 </td> <td>600 </td> <td>120 </td> <td>4 </td> <td>30 </td> <td>3 </td> <td>5 </td> <td><a href="/wiki/Triangular_bipyramid" title="Triangular bipyramid">Triangular</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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 {5}}-1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {{\sqrt {5}}-1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef1a9c193af6a32c6c4e27480e8fab42240028c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.128ex; height:4.343ex;" alt="{\textstyle {\frac {{\sqrt {5}}-1}{3}}}"></span> </td> <td>0.613 </td> <td><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 {10+9{\sqrt {5}}}{61}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mo>+</mo> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>61</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {10+9{\sqrt {5}}}{61}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/396490c6656a0835d47e43e16247ed38b09f94ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.58ex; height:4.343ex;" alt="{\textstyle -{\frac {10+9{\sqrt {5}}}{61}}}"></span> </td> <td><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 {7-12{\sqrt {5}}}{61}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mo>−</mo> <mn>12</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>61</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {7-12{\sqrt {5}}}{61}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3ac24a708a0c4cad1263be967cb29f4ea76c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.58ex; height:4.343ex;" alt="{\textstyle -{\frac {7-12{\sqrt {5}}}{61}}}"></span> </td></tr> <tr align="center"> <td><a href="/wiki/24-cell" title="24-cell">R. 16-cell</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>24 <sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </td> <td>8 </td> <td>16 </td> <td>6 </td> <td>6 </td> <td>3 </td> <td>3 </td> <td><a href="/wiki/Square_bipyramid" class="mw-redirect" title="Square bipyramid">Square</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5094a5b1e2f42490aa4de2c7a4b7235a27f1b73f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\textstyle {\sqrt {2}}}"></span> </td> <td>1 </td> <td><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}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70def1351375878dbed3e58e789193c219f2347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {1}{3}}}"></span> </td> <td><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}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70def1351375878dbed3e58e789193c219f2347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\textstyle -{\frac {1}{3}}}"></span> </td></tr> <tr align="center"> <td><a href="/wiki/Rectified_cubic_honeycomb" class="mw-redirect" title="Rectified cubic honeycomb">R. cubic <br /> honeycomb</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>∞ </td> <td>∞ </td> <td>∞ </td> <td>6 </td> <td>12 </td> <td>3 </td> <td>4 </td> <td><a href="/wiki/Square_bipyramid" class="mw-redirect" title="Square bipyramid">Square</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6706df9ed9f240d1a94545fb4e522bda168fe8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\textstyle 1}"></span> </td> <td>0.866 </td> <td><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d70cb520e8614d09bcb868a9ec8559f5b25d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\textstyle -{\frac {1}{2}}}"></span> </td> <td><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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f408e516b4d3056afd68301d502498822d97b9be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\textstyle 0}"></span> </td></tr> <tr align="center"> <td><a href="/wiki/Rectified_600-cell" title="Rectified 600-cell">R. 600-cell</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>720 </td> <td>120 </td> <td>600 </td> <td>12 </td> <td>6 </td> <td>3 </td> <td>3 </td> <td><a href="/wiki/Pentagonal_bipyramid" title="Pentagonal bipyramid">Pentagonal</a> </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1c/CDel_2x.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td><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 {5+3{\sqrt {5}}}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {5+3{\sqrt {5}}}{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e40906134d82a809563e23dd5150422698cb2f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.949ex; height:4.343ex;" alt="{\textstyle {\frac {5+3{\sqrt {5}}}{5}}}"></span> </td> <td>1.447 </td> <td><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 {11+4{\sqrt {5}}}{41}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>41</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6e848702bc3e97ff4d67049b9710e4a772d714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.58ex; height:4.176ex;" alt="{\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}"></span> </td> <td><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 {11+4{\sqrt {5}}}{41}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>41</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6e848702bc3e97ff4d67049b9710e4a772d714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.58ex; height:4.176ex;" alt="{\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Other_dimensions">Other dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=9" title="Edit section: Other dimensions"><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:Romb_deltoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Romb_deltoid.svg/180px-Romb_deltoid.svg.png" decoding="async" width="180" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Romb_deltoid.svg/270px-Romb_deltoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Romb_deltoid.svg/360px-Romb_deltoid.svg.png 2x" data-file-width="138" data-file-height="104" /></a><figcaption>A rhombus is a 2-dimensional analog of a right symmetric bipyramid</figcaption></figure> <p>A generalized <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional "bipyramid" is any <span class="texhtml mvar" style="font-style:italic;">n</span>-<a href="/wiki/Polytope" title="Polytope">polytope</a> constructed from an <span class="texhtml">(<i>n</i> − 1)</span>-polytope <i>base</i> lying in a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a>, with every base vertex connected by an edge to two <i>apex</i> vertices. If the <span class="texhtml">(<i>n</i> − 1)</span>-polytope is a regular polytope and the apices are equidistant from its center along the line perpendicular to the base hyperplane, it will have identical <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramidal</a> <a href="/wiki/Facet_(geometry)" title="Facet (geometry)">facets</a>. </p><p>A 2-dimensional analog of a right symmetric bipyramid is formed by joining two <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> <a href="/wiki/Isosceles_triangle" title="Isosceles triangle">isosceles triangles</a> base-to-base to form a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a>. More generally, a <a href="/wiki/Kite_(geometry)" title="Kite (geometry)">kite</a> is a 2-dimensional analog of a (possibly asymmetric) right bipyramid, and any quadrilateral is a 2-dimensional analog of a general bipyramid. </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=Bipyramid&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Trapezohedron" title="Trapezohedron">Trapezohedron</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-right_pyramids-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-right_pyramids_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-right_pyramids_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">The center of a regular polygon is unambiguous, but for irregular polygons sources disagree. Some sources only allow a right pyramid to have a regular polygon as a base. Others define a right pyramid as having its apices on a line perpendicular to the base and passing through its <a href="/wiki/Centroid" title="Centroid">centroid</a>. <a href="#CITEREFPolya1954">Polya (1954)</a> restricts right pyramids to those with a <a href="/wiki/Tangential_polygon" title="Tangential polygon">tangential polygon</a> for a base, with the apices on a line perpendicular to the base and passing through the <a href="/wiki/Incenter" title="Incenter">incenter</a>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">The smallest geometric di-<span class="texhtml mvar" style="font-style:italic;">n</span>-gonal bipyramids have eight faces, and are topologically identical to the regular <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>. In this case (<span class="texhtml">2<i>n</i> = 2×2</span>):<br />an isotoxal right (symmetric) didigonal bipyramid is called a <i>rhombic bipyramid</i>,<sup id="cite_ref-tulane_14-1" class="reference"><a href="#cite_note-tulane-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-uwgb_17-0" class="reference"><a href="#cite_note-uwgb-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> although all its faces are scalene triangles, because its flat polygon base is a rhombus.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Given numerically due to more complex form.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">The rectified 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=12" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-aarts-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-aarts_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-aarts_2-1"><sup><i><b>b</b></i></sup></a></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="CITEREFAarts2008" class="citation book cs1">Aarts, J. M. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1ctGAAAAQBAJ&pg=PA303"><i>Plane and Solid Geometry</i></a>. Springer. p. 303. <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%2F978-0-387-78241-6">10.1007/978-0-387-78241-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-78241-6" title="Special:BookSources/978-0-387-78241-6"><bdi>978-0-387-78241-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plane+and+Solid+Geometry&rft.pages=303&rft.pub=Springer&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-0-387-78241-6&rft.isbn=978-0-387-78241-6&rft.aulast=Aarts&rft.aufirst=J.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1ctGAAAAQBAJ%26pg%3DPA303&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-polya-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-polya_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolya1954" class="citation book cs1">Polya, G. (1954). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-TWTcSa19jkC&pg=PA138"><i>Mathematics and Plausible Reasoning: Induction and analogy in mathematics</i></a>. Princeton University Press. p. 138. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-02509-6" title="Special:BookSources/0-691-02509-6"><bdi>0-691-02509-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Plausible+Reasoning%3A+Induction+and+analogy+in+mathematics&rft.pages=138&rft.pub=Princeton+University+Press&rft.date=1954&rft.isbn=0-691-02509-6&rft.aulast=Polya&rft.aufirst=G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-TWTcSa19jkC%26pg%3DPA138&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-montroll-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-montroll_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMontroll2009" class="citation book cs1"><a href="/wiki/John_Montroll" title="John Montroll">Montroll, John</a> (2009). <a href="/wiki/Origami_Polyhedra_Design" title="Origami Polyhedra Design"><i>Origami Polyhedra Design</i></a>. A K Peters. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SeTqBgAAQBAJ&pg=PA6">p. 6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781439871065" title="Special:BookSources/9781439871065"><bdi>9781439871065</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Origami+Polyhedra+Design&rft.pages=p.+6&rft.pub=A+K+Peters&rft.date=2009&rft.isbn=9781439871065&rft.aulast=Montroll&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-trigg-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-trigg_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrigg1978" class="citation journal cs1">Trigg, Charles W. (1978). "An infinite class of deltahedra". <i>Mathematics Magazine</i>. <b>51</b> (1): 55–57. <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%2F0025570X.1978.11976675">10.1080/0025570X.1978.11976675</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/2689647">2689647</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=1572246">1572246</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=An+infinite+class+of+deltahedra&rft.volume=51&rft.issue=1&rft.pages=55-57&rft.date=1978&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1572246%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2689647%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F0025570X.1978.11976675&rft.aulast=Trigg&rft.aufirst=Charles+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-uehara-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-uehara_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUehara2020" class="citation book cs1">Uehara, Ryuhei (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62"><i>Introduction to Computational Origami: The World of New Computational Geometry</i></a>. Springer. p. 62. <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%2F978-981-15-4470-5">10.1007/978-981-15-4470-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-15-4470-5" title="Special:BookSources/978-981-15-4470-5"><bdi>978-981-15-4470-5</bdi></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:220150682">220150682</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+Computational+Origami%3A+The+World+of+New+Computational+Geometry&rft.pages=62&rft.pub=Springer&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220150682%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-981-15-4470-5&rft.isbn=978-981-15-4470-5&rft.aulast=Uehara&rft.aufirst=Ryuhei&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D51juDwAAQBAJ%26pg%3DPA62&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-cromwell-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-cromwell_7-0">^</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. 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"Dungeons, dragons, and dice". <i>The Mathematical Gazette</i>. <b>74</b> (469): 243–256. <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=243-256&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%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-sibley-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-sibley_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSibley2015" class="citation book cs1">Sibley, Thomas Q. (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EUh2CgAAQBAJ&pg=PA53"><i>Thinking Geometrically: A Survey of Geometries</i></a>. Mathematical Association of American. p. 53. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-939512-08-6" title="Special:BookSources/978-1-939512-08-6"><bdi>978-1-939512-08-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thinking+Geometrically%3A+A+Survey+of+Geometries&rft.pages=53&rft.pub=Mathematical+Association+of+American&rft.date=2015&rft.isbn=978-1-939512-08-6&rft.aulast=Sibley&rft.aufirst=Thomas+Q.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEUh2CgAAQBAJ%26pg%3DPA53&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-king-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-king_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKing1994" class="citation book cs1">King, Robert B. (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c3fsCAAAQBAJ&pg=PA113">"Polyhedral Dynamics"</a>. In Bonchev, Danail D.; Mekenyan, O.G. (eds.). <i>Graph Theoretical Approaches to Chemical Reactivity</i>. Springer. <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%2F978-94-011-1202-4">10.1007/978-94-011-1202-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-011-1202-4" title="Special:BookSources/978-94-011-1202-4"><bdi>978-94-011-1202-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Polyhedral+Dynamics&rft.btitle=Graph+Theoretical+Approaches+to+Chemical+Reactivity&rft.pub=Springer&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2F978-94-011-1202-4&rft.isbn=978-94-011-1202-4&rft.aulast=King&rft.aufirst=Robert+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc3fsCAAAQBAJ%26pg%3DPA113&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-armstrong-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-armstrong_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArmstrong1988" class="citation book cs1">Armstrong, M. A. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f2AFCAAAQBAJ&pg=PA39"><i>Group and Symmetry</i></a>. Undergraduate Texts in Mathematics. Springer. p. 39. <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%2F978-1-4757-4034-9">10.1007/978-1-4757-4034-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-4034-9" title="Special:BookSources/978-1-4757-4034-9"><bdi>978-1-4757-4034-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+and+Symmetry&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=39&rft.pub=Springer&rft.date=1988&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-4034-9&rft.isbn=978-1-4757-4034-9&rft.aulast=Armstrong&rft.aufirst=M.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df2AFCAAAQBAJ%26pg%3DPA39&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-tulane-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-tulane_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-tulane_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-tulane_14-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-tulane_14-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-tulane_14-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-tulane_14-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm">"Crystal Form, Zones, Crystal Habit"</a>. <i>Tulane.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">16 September</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Tulane.edu&rft.atitle=Crystal+Form%2C+Zones%2C+Crystal+Habit&rft_id=http%3A%2F%2Fwww.tulane.edu%2F~sanelson%2Feens211%2Fforms_zones_habit.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_ditrigonal_bipyramidal_class,_p._581_(p._603_on_Wikisource)-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_ditrigonal_bipyramidal_class,_p._581_(p._603_on_Wikisource)_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSpencer1911">Spencer 1911</a>, 6. Hexagonal system, <i>rhombohedral division</i>, ditrigonal bipyramidal class, p. 581 (p. 603 on Wikisource).</span> </li> <li id="cite_note-FOOTNOTESpencer19112._Tegragonal_system,_holosymmetric_class,_fig._46,_p._577_(p._599_on_Wikisource)-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESpencer19112._Tegragonal_system,_holosymmetric_class,_fig._46,_p._577_(p._599_on_Wikisource)_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSpencer1911">Spencer 1911</a>, 2. Tegragonal system, holosymmetric class, fig. 46, p. 577 (p. 599 on Wikisource).</span> </li> <li id="cite_note-uwgb-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-uwgb_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-uwgb_17-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-uwgb_17-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-uwgb_17-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-uwgb_17-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130918103121/https://www.uwgb.edu/dutchs/symmetry/xlforms.htm">"The 48 Special Crystal Forms"</a>. 18 September 2013. Archived from <a rel="nofollow" class="external text" href="https://www.uwgb.edu/dutchs/symmetry/xlforms.htm">the original</a> on 18 September 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">18 November</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+48+Special+Crystal+Forms&rft.date=2013-09-18&rft_id=https%3A%2F%2Fwww.uwgb.edu%2Fdutchs%2Fsymmetry%2Fxlforms.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-kp-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-kp_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleinPhilpotts2013" class="citation book cs1">Klein, Cornelis; Philpotts, Anthony R. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V7nUnYKmrxgC&pg=PA108"><i>Earth Materials: Introduction to Mineralogy and Petrology</i></a>. Cambridge University Press. p. 108. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-14521-3" title="Special:BookSources/978-0-521-14521-3"><bdi>978-0-521-14521-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Earth+Materials%3A+Introduction+to+Mineralogy+and+Petrology&rft.pages=108&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-0-521-14521-3&rft.aulast=Klein&rft.aufirst=Cornelis&rft.au=Philpotts%2C+Anthony+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DV7nUnYKmrxgC%26pg%3DPA108&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_holosymmetric_class,_fig._68,_p._580_(p._602_on_Wikisource)-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESpencer19116._Hexagonal_system,_''rhombohedral_division'',_holosymmetric_class,_fig._68,_p._580_(p._602_on_Wikisource)_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSpencer1911">Spencer 1911</a>, 6. Hexagonal system, <i>rhombohedral division</i>, holosymmetric class, fig. 68, p. 580 (p. 602 on Wikisource).</span> </li> <li id="cite_note-FOOTNOTESpencer19112._Tetragonal_system,_scalenohedral_class,_fig._51,_p._577_(p._599_on_Wikisource)-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESpencer19112._Tetragonal_system,_scalenohedral_class,_fig._51,_p._577_(p._599_on_Wikisource)_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSpencer1911">Spencer 1911</a>, p. 2. Tetragonal system, scalenohedral class, fig. 51, p. 577 (p. 599 on Wikisource).</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRankin1988" class="citation journal cs1">Rankin, John R. (1988). "Classes of polyhedra defined by jet graphics". <i>Computers & Graphics</i>. <b>12</b> (2): 239–254. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0097-8493%2888%2990036-2">10.1016/0097-8493(88)90036-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+%26+Graphics&rft.atitle=Classes+of+polyhedra+defined+by+jet+graphics&rft.volume=12&rft.issue=2&rft.pages=239-254&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0097-8493%2888%2990036-2&rft.aulast=Rankin&rft.aufirst=John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Works_Cited">Works Cited</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=13" title="Edit section: Works Cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnthony_Pugh1976" class="citation book cs1">Anthony Pugh (1976). <i>Polyhedra: A visual approach</i>. California: University of California Press Berkeley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-520-03056-7" title="Special:BookSources/0-520-03056-7"><bdi>0-520-03056-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyhedra%3A+A+visual+approach&rft.place=California&rft.pub=University+of+California+Press+Berkeley&rft.date=1976&rft.isbn=0-520-03056-7&rft.au=Anthony+Pugh&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span> Chapter 4: Duals of the Archimedean polyhedra, prisms and antiprisms</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpencer1911" class="citation encyclopaedia cs1"><a href="/wiki/Leonard_James_Spencer" title="Leonard James Spencer">Spencer, Leonard James</a> (1911). <span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Crystallography"><a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Crystallography">"Crystallography" </a></span>. In <a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a> (ed.). <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. Vol. 07 (11th ed.). Cambridge University Press. pp. 569–591.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Crystallography&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=569-591&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rft.aulast=Spencer&rft.aufirst=Leonard+James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bipyramid&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Bipyramids" class="extiw" title="commons:Category:Bipyramids">Bipyramids</a></span>.</div></div> </div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Dipyramid"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Dipyramid.html">"Dipyramid"</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=Dipyramid&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDipyramid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Isohedron"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Isohedron.html">"Isohedron"</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=Isohedron&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FIsohedron.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABipyramid" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathconsult.ch/showroom/unipoly/">The Uniform Polyhedra</a></li> <li><a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> The Encyclopedia of Polyhedra <ul><li><a href="/wiki/VRML" title="VRML">VRML</a> models <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/alphabetic-list.html">(George Hart)</a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/triangular_dipyramid.wrl"><3></a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/octahedron.wrl"><4></a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_dipyramid.wrl"><5></a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_dipyramid.wrl"><6></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150203062131/http://www.georgehart.com/virtual-polyhedra/vrml/heptagonal_dipyramid.wrl"><7></a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/octagonal_dipyramid.wrl"><8></a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/enneagonal_dipyramid.wrl"><9></a> <a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vrml/decagonal_dipyramid.wrl"><10></a> <ul><li><a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/conway_notation.html">Conway Notation for Polyhedra</a> Try: "dP<i>n</i>", where <i>n</i> = 3, 4, 5, 6, ... 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aria-labelledby="Convex_polyhedra" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><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 a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Convex_polyhedron_navigator" title="Template:Convex polyhedron navigator"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Convex_polyhedron_navigator" title="Template talk:Convex polyhedron navigator"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Convex_polyhedron_navigator" title="Special:EditPage/Template:Convex polyhedron navigator"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Convex_polyhedra" style="font-size:114%;margin:0 4em">Convex <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> <span class="nobold">(<a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedron#Regular_tetrahedron" title="Tetrahedron">tetrahedron</a></li> <li><a href="/wiki/Cube" title="Cube">cube</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">octahedron</a></li> <li><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a></li> <li><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a><br /><span class="nobold">(<a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li> <li><a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li> <li><a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a></li> <li><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li> <li><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li> <li><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li> <li><a href="/wiki/Snub_cube" title="Snub cube">snub cube</a></li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li> <li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li> <li><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li> <li><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a><br /><span class="nobold">(duals of Archimedean)</span></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li> <li><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li> <li><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li> <li><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li> <li><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li> <li><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li> <li><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li> <li><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Dihedron" title="Dihedron">dihedron</a></i></li> <li><i><a href="/wiki/Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Bipyramid&oldid=1253509482">https://en.wikipedia.org/w/index.php?title=Bipyramid&oldid=1253509482</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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Bipyramid - Wikipedia