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</li> <li id="toc-Structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structure"> <div class="vector-toc-text"> 1.2 Structure </div> </a> <ul id="toc-Structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotations"> <div class="vector-toc-text"> 1.3 Rotations </div> </a> <ul id="toc-Rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> 1.4 Constructions </div> </a> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Octahedral_dipyramid" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Octahedral_dipyramid"> <div class="vector-toc-text"> 1.4.1 Octahedral dipyramid </div> </a> <ul id="toc-Octahedral_dipyramid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tetrahedral_constructions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tetrahedral_constructions"> <div class="vector-toc-text"> 1.4.2 Tetrahedral constructions </div> </a> <ul id="toc-Tetrahedral_constructions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Helical_construction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Helical_construction"> <div class="vector-toc-text"> 1.4.3 Helical construction </div> </a> <ul id="toc-Helical_construction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-As_a_configuration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_configuration"> <div class="vector-toc-text"> 1.5 As a configuration </div> </a> <ul id="toc-As_a_configuration-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tessellations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tessellations"> <div class="vector-toc-text"> 2 Tessellations </div> </a> <ul id="toc-Tessellations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projections" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Projections"> <div class="vector-toc-text"> 3 Projections </div> </a> <ul id="toc-Projections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4_sphere_Venn_diagram" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#4_sphere_Venn_diagram"> <div class="vector-toc-text"> 4 4 sphere Venn diagram </div> </a> <ul id="toc-4_sphere_Venn_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry_constructions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symmetry_constructions"> <div class="vector-toc-text"> 5 Symmetry constructions </div> </a> <ul id="toc-Symmetry_constructions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_complex_polygons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_complex_polygons"> <div class="vector-toc-text"> 6 Related complex polygons </div> </a> <ul id="toc-Related_complex_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_uniform_polytopes_and_honeycombs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_uniform_polytopes_and_honeycombs"> <div class="vector-toc-text"> 7 Related uniform polytopes and honeycombs </div> </a> <ul id="toc-Related_uniform_polytopes_and_honeycombs-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"> 8 See also </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"> 9 Notes </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"> 10 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 11 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item 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Czech" lang="cs" hreflang="cs" data-title="16nadstěn" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Hexadecacoron" title="Hexadecacoron – Spanish" lang="es" hreflang="es" data-title="Hexadecacoron" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/16-%C4%89elo" title="16-ĉelo – Esperanto" lang="eo" hreflang="eo" data-title="16-ĉelo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Hexad%C3%A9cachore" title="Hexadécachore – French" lang="fr" hreflang="fr" data-title="Hexadécachore" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%8B%AD%EC%9C%A1%ED%8F%AC%EC%B2%B4" title="정십육포체 – Korean" lang="ko" hreflang="ko" data-title="정십육포체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Esadecacoro" title="Esadecacoro – Italian" lang="it" hreflang="it" data-title="Esadecacoro" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AD%A3%E5%8D%81%E5%85%AD%E8%83%9E%E4%BD%93" title="正十六胞体 – Japanese" lang="ja" hreflang="ja" data-title="正十六胞体" data-language-autonym="日本語" 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(May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">16-cell (4-orthoplex)</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Schlegel_wireframe_16-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/240px-Schlegel_wireframe_16-cell.png" decoding="async" width="240" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/360px-Schlegel_wireframe_16-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/480px-Schlegel_wireframe_16-cell.png 2x" data-file-width="768" data-file-height="786" /></a><div class="infobox-caption"><a href="/wiki/Schlegel_diagram" title="Schlegel diagram">Schlegel diagram</a> (vertices and edges)</div></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Convex_regular_4-polytope" class="mw-redirect" title="Convex regular 4-polytope">Convex regular 4-polytope</a> 4-<a href="/wiki/Orthoplex" class="mw-redirect" title="Orthoplex">orthoplex</a> 4-<a href="/wiki/Demihypercube" title="Demihypercube">demicube</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a></th><td class="infobox-data">{3,3,4}</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagram</a></th><td class="infobox-data"><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/3-face" class="mw-redirect" title="3-face">Cells</a></th><td class="infobox-data">16 <a href="/wiki/Tetrahedron" title="Tetrahedron">{3,3}</a> <a href="/wiki/File:3-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/3-simplex_t0.svg/25px-3-simplex_t0.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/3-simplex_t0.svg/38px-3-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/3-simplex_t0.svg/50px-3-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/2-face" class="mw-redirect" title="2-face">Faces</a></th><td class="infobox-data">32 <a href="/wiki/Triangle" title="Triangle">{3}</a> <a href="/wiki/File:2-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/2-simplex_t0.svg/25px-2-simplex_t0.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/2-simplex_t0.svg/38px-2-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/2-simplex_t0.svg/50px-2-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Edge_(geometry)" title="Edge (geometry)">Edges</a></th><td class="infobox-data">24</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></th><td class="infobox-data">8</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Vertex_figure" title="Vertex figure">Vertex figure</a></th><td class="infobox-data"><a href="/wiki/File:16-cell_verf.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/16-cell_verf.svg/80px-16-cell_verf.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/16-cell_verf.svg/120px-16-cell_verf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/16-cell_verf.svg/160px-16-cell_verf.svg.png 2x" data-file-width="240" data-file-height="240" /></a> <a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygon</a></th><td class="infobox-data"><a href="/wiki/Octagon" title="Octagon">octagon</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a></th><td class="infobox-data">B4, [3,3,4], order 384 D4, order 192</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dual_polytope" class="mw-redirect" title="Dual polytope">Dual</a></th><td class="infobox-data"><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="/wiki/Convex_polytope" title="Convex polytope">convex</a>, <a href="/wiki/Isogonal_figure" title="Isogonal figure">isogonal</a>, <a href="/wiki/Isotoxal_figure" title="Isotoxal figure">isotoxal</a>, <a href="/wiki/Isohedral_figure" title="Isohedral figure">isohedral</a>, <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a>, <a href="/wiki/Hanner_polytope" title="Hanner polytope">Hanner polytope</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Uniform_4-polytope#Convex_uniform_4-polytopes" title="Uniform 4-polytope">Uniform index</a></th><td class="infobox-data">12</td></tr></tbody></table> In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the 16-cell is the <a href="/wiki/Regular_convex_4-polytope" class="mw-redirect" title="Regular convex 4-polytope">regular convex 4-polytope</a> (four-dimensional analogue of a Platonic solid) with <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician <a href="/wiki/Ludwig_Schl%C3%A4fli" title="Ludwig Schläfli">Ludwig Schläfli</a> in the mid-19th century.<a href="#cite_note-FOOTNOTECoxeter1973141§&nbsp;7-x._Historical_remarks-1">[1]</a> It is also called C16, hexadecachoron,<a href="#cite_note-2">[2]</a> or hexdecahedroid [<a href="/wiki/Sic" title="Sic">sic</a>?] .<a href="#cite_note-3">[3]</a> It is the 4-dimesional member of an infinite family of polytopes called <a href="/wiki/Cross-polytope" title="Cross-polytope">cross-polytopes</a>, orthoplexes, or hyperoctahedrons which are analogous to the <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> in three dimensions. It is Coxeter's <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16fb7d2f7d9b9f310820b4e110b084003aa80fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{4}}"> polytope.B_4-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973120=121§&nbsp;7.2._See_illustration_Fig_7.2B-4">[4]</a> The <a href="/wiki/Dual_polytope" class="mw-redirect" title="Dual polytope">dual polytope</a> is the <a href="/wiki/Tesseract" title="Tesseract">tesseract</a> (4-<a href="/wiki/Hypercube" title="Hypercube">cube</a>), which it can be combined with to form a compound figure. The cells of the 16-cell are dual to the 16 vertices of the tesseract. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=1" title="Edit section: Geometry">edit</a>]</div> The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).<a href="#cite_note-polytopes_ordered_by_size_and_complexity-6">[a]</a> Each of its 4 successor convex regular 4-polytopes can be constructed as the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of a <a href="/wiki/Polytope_compound" title="Polytope compound">polytope compound</a> of multiple 16-cells: the 16-vertex <a href="/wiki/Tesseract" title="Tesseract">tesseract</a> as a compound of two 16-cells, the 24-vertex <a href="/wiki/24-cell" title="24-cell">24-cell</a> as a compound of three 16-cells, the 120-vertex <a href="/wiki/600-cell" title="600-cell">600-cell</a> as a compound of fifteen 16-cells, and the 600-vertex <a href="/wiki/120-cell" title="120-cell">120-cell</a> as a compound of seventy-five 16-cells.<a href="#cite_note-7">[b]</a> <table class="wikitable mw-collapsible mw-collapsed" style="white-space:nowrap;text-align:center;"> <tbody><tr> <th colspan="8"><a href="/wiki/Regular_4-polytopes" class="mw-redirect" title="Regular 4-polytopes">Regular convex 4-polytopes</a> </th></tr> <tr> <th style="text-align:right;"><a href="/wiki/Coxeter_group" title="Coxeter group">Symmetry group</a> </th> <td><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">A4</a> </td> <td colspan="2"><a href="/wiki/Hyperoctahedral_group" title="Hyperoctahedral group">B4</a> </td> <td><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a> </td> <td colspan="2"><a href="/wiki/H4_polytope" title="H4 polytope">H4</a> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Name </th> <td style="vertical-align:top;"><a href="/wiki/5-cell" title="5-cell">5-cell</a> Hyper-<a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> 5-point </td> <td style="vertical-align:top;"><a class="mw-selflink selflink">16-cell</a> Hyper-<a href="/wiki/Octahedron" title="Octahedron">octahedron</a> 8-point </td> <td style="vertical-align:top;"><a href="/wiki/8-cell" class="mw-redirect" title="8-cell">8-cell</a> Hyper-<a href="/wiki/Cube" title="Cube">cube</a> 16-point </td> <td style="vertical-align:top;"><a href="/wiki/24-cell" title="24-cell">24-cell</a> 24-point </td> <td style="vertical-align:top;"><a href="/wiki/600-cell" title="600-cell">600-cell</a> Hyper-<a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a> 120-point </td> <td style="vertical-align:top;"><a href="/wiki/120-cell" title="120-cell">120-cell</a> Hyper-<a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a> 600-point </td></tr> <tr> <th style="text-align:right;"><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <td>{3, 3, 3} </td> <td>{3, 3, 4} </td> <td>{4, 3, 3} </td> <td>{3, 4, 3} </td> <td>{3, 3, 5} </td> <td>{5, 3, 3} </td></tr> <tr> <th style="text-align:right;"><a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter mirrors</a> </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td></tr> <tr> <th style="text-align:right;">Mirror dihedrals </th> <td><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/4⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/4⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/4⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/5⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/5⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/3⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠𝝅/2⁠ </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Graph </th> <td><a href="/wiki/File:4-simplex_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/120px-4-simplex_t0.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/180px-4-simplex_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/240px-4-simplex_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:4-cube_t3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/120px-4-cube_t3.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/180px-4-cube_t3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/240px-4-cube_t3.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:4-cube_t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/120px-4-cube_t0.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/180px-4-cube_t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/240px-4-cube_t0.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:24-cell_t0_F4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/120px-24-cell_t0_F4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/180px-24-cell_t0_F4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/240px-24-cell_t0_F4.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:600-cell_graph_H4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/120px-600-cell_graph_H4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/180px-600-cell_graph_H4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/240px-600-cell_graph_H4.svg.png 2x" data-file-width="800" data-file-height="800" /></a> </td> <td><a href="/wiki/File:120-cell_graph_H4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/120px-120-cell_graph_H4.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/180px-120-cell_graph_H4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/240px-120-cell_graph_H4.svg.png 2x" data-file-width="800" data-file-height="800" /></a> </td></tr> <tr> <th style="text-align:right;">Vertices </th> <td>5 tetrahedral </td> <td>8 octahedral </td> <td>16 tetrahedral </td> <td>24 cubical </td> <td>120 icosahedral </td> <td>600 tetrahedral </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/120-cell#Chords" title="120-cell">Edges</a> </th> <td>10 triangular </td> <td>24 square </td> <td>32 triangular </td> <td>96 triangular </td> <td>720 pentagonal </td> <td>1200 triangular </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Faces </th> <td>10 triangles </td> <td>32 triangles </td> <td>24 squares </td> <td>96 triangles </td> <td>1200 triangles </td> <td>720 pentagons </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Cells </th> <td>5 tetrahedra </td> <td>16 tetrahedra </td> <td>8 cubes </td> <td>24 octahedra </td> <td>600 tetrahedra </td> <td>120 dodecahedra </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/600-cell#Clifford_parallel_cell_rings" title="600-cell">Tori</a> </th> <td>1 <a href="/wiki/5-cell#Boerdijk–Coxeter_helix" title="5-cell">5-tetrahedron</a> </td> <td>2 <a class="mw-selflink-fragment" href="#Helical_construction">8-tetrahedron</a> </td> <td>2 <a href="/wiki/8-cell#Construction" class="mw-redirect" title="8-cell">4-cube</a> </td> <td>4 <a href="/wiki/24-cell#Cell_rings" title="24-cell">6-octahedron</a> </td> <td>20 <a href="/wiki/600-cell#Boerdijk–Coxeter_helix_rings" title="600-cell">30-tetrahedron</a> </td> <td>12 <a href="/wiki/120-cell#Intertwining_rings" title="120-cell">10-dodecahedron</a> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Inscribed </th> <td>120 in 120-cell </td> <td>675 in 120-cell </td> <td>2 16-cells </td> <td>3 8-cells </td> <td>25 24-cells </td> <td>10 600-cells </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/Great_circle" title="Great circle">Great polygons</a> </th> <td> </td> <td>2 <a class="mw-selflink-fragment" href="#Coordinates">squares</a> x 3 </td> <td>4 rectangles x 4 </td> <td>4 <a href="/wiki/24-cell#Hexagons" title="24-cell">hexagons</a> x 4 </td> <td>12 <a href="/wiki/600-cell#Geodesics" title="600-cell">decagons</a> x 6 </td> <td>100 <a href="/wiki/120-cell#Chords" title="120-cell">irregular hexagons</a> x 4 </td></tr> <tr> <th style="vertical-align:top;text-align:right;"><a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygons</a> </th> <td>1 <a href="/wiki/5-cell#Boerdijk–Coxeter_helix" title="5-cell">pentagon</a> x 2 </td> <td>1 <a class="mw-selflink-fragment" href="#Helical_construction">octagon</a> x 3 </td> <td>2 <a href="/wiki/Octagon#Skew_octagon" title="Octagon">octagons</a> x 4 </td> <td>2 <a href="/wiki/Dodecagon#Skew_dodecagon" title="Dodecagon">dodecagons</a> x 4 </td> <td>4 <a href="/wiki/30-gon#Petrie_polygons" class="mw-redirect" title="30-gon">30-gons</a> x 6 </td> <td>20 <a href="/wiki/30-gon#Petrie_polygons" class="mw-redirect" title="30-gon">30-gons</a> x 4 </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Long radius </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Edge length </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>1.581</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7891ac9b4af750cc3e54f5f21486f4b396729a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\approx 1.414}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.414</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1.414}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fafaf1870e0527a631858dcc0c7579b34ff297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.493ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\approx 1.414}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>ϕ</mi> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.618</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1e8a92bb277903a43ca3ae567ea64505fc32eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.211ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.270</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29afe3cfc54031633a59df0637d1a3f7f71bf684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.233ex; height:4.343ex;" alt="{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}"> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Short radius </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4825cd2a1ca51dfc4d53042434f6d3733370a57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.707</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4dc0f6527d5ac3f56ed3e68bc66e0b2f6a2bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.926</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05c35ecaf7f1d828adb58f78720535d06a3c177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.366ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.926</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05c35ecaf7f1d828adb58f78720535d06a3c177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.366ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}"> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Area </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>10.825</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaa7c7d04842630db48b33b9d74c338be70c6b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.894ex; height:4.843ex;" alt="{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>32</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>27.713</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b95c593579e89ea2c8e3ebb2207bed8433a02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.027ex; height:4.843ex;" alt="{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 24}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be92101c8b0277e66fdefeef1ccdd7788e88ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 24}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>96</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>16</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>41.569</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b99126f57cc5b6592e4ef72670abbdf84068c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.849ex; height:4.843ex;" alt="{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mrow> <mn>4</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>198.48</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5688be271d95c5759354e1e5459c0fc1f8cbfc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.839ex; height:5.009ex;" alt="{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>720</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>25</mn> <mo>+</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </msqrt> <mrow> <mn>8</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>90.366</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14322accb342db5a422b3b14329f5e4576ef6e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.35ex; height:7.509ex;" alt="{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}"> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">Volume </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>24</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>2.329</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8253a425495bc8bc6c197e38e71d1eb1515a37d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.569ex; height:4.843ex;" alt="{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>5.333</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4c8bcc6a19368c79cc7db6ebad8bb5b10d5cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.541ex; height:4.843ex;" alt="{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5" 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="{\displaystyle 8}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>24</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>11.314</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd86a479242f86673a77d371d02474fd1211994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.072ex; height:4.843ex;" alt="{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>600</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mrow> <mn>12</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>16.693</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6510bf43702967fc6602d15435caea9989169b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.499ex; height:5.009ex;" alt="{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>120</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>15</mn> <mo>+</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> </msqrt> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mn>18.118</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e90aaf6322085e2c0dd2bf8ca33e054d4c46592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.979ex; height:5.009ex;" alt="{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}"> </td></tr> <tr> <th style="vertical-align:top;text-align:right;">4-Content </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>24</mn> </mfrac> </mstyle> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>0.146</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd8259f9d61b844367d627adf4da0edae4aef88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.279ex; height:5.343ex;" alt="{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2}{3}}\approx 0.667}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>0.667</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}\approx 0.667}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f729b57d66245544a3f57b8d2b0a31a0a9a883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.053ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}\approx 0.667}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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="{\displaystyle 2}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Short</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Vol</mtext> </mrow> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>3.863</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04209037c0378ec46376aba5e33615134ace5ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.956ex; height:3.676ex;" alt="{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Short</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Vol</mtext> </mrow> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>4.193</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f33beed2b51250ff3c61ead8b4214092c74e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.956ex; height:3.676ex;" alt="{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Coordinates">Coordinates</h3>[<a href="/w/index.php?title=16-cell&action=edit&section=2" title="Edit section: Coordinates">edit</a>]</div> <table class="wikitable floatright"> <tbody><tr> <th colspan="2">Disjoint squares </th></tr> <tr> <td> <table class="wikitable" style="white-space:nowrap;"> <tbody><tr> <th colspan="2">xy plane </th></tr> <tr> <td>( 0, 1, 0, 0)</td> <td>( 0, 0,-1, 0) </td></tr> <tr> <td>( 0, 0, 1, 0)</td> <td>( 0,-1, 0, 0) </td></tr></tbody></table> </td></tr> <tr> <td> <table class="wikitable" style="white-space:nowrap;"> <tbody><tr> <th colspan="2">wz plane </th></tr> <tr> <td>( 1, 0, 0, 0)</td> <td>( 0, 0, 0,-1) </td></tr> <tr> <td>( 0, 0, 0, 1)</td> <td>(-1, 0, 0, 0) </td></tr></tbody></table> </td></tr></tbody></table>The 16-cell is the 4-dimensional <a href="/wiki/Cross_polytope" class="mw-redirect" title="Cross polytope">cross polytope (4-orthoplex)</a>, which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system. The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is √2. The vertex coordinates form 6 <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> central squares lying in the 6 coordinate planes. Squares in opposite planes that do not share an axis (e.g. in the xy and wz planes) are completely disjoint (they do not intersect at any vertices).<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> The 16-cell constitutes an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes. <div class="mw-heading mw-heading3"><h3 id="Structure">Structure</h3>[<a href="/w/index.php?title=16-cell&action=edit&section=3" title="Edit section: Structure">edit</a>]</div> The <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> of the 16-cell is {3,3,4}, indicating that its cells are <a href="/wiki/Regular_tetrahedron" class="mw-redirect" title="Regular tetrahedron">regular tetrahedra</a> {3,3} and its <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> is a <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a> {3,4}. There are 8 tetrahedra, 12 triangles, and 6 edges meeting at every vertex. Its <a href="/wiki/Edge_figure" class="mw-redirect" title="Edge figure">edge figure</a> is a square. There are 4 tetrahedra and 4 triangles meeting at every edge. The 16-cell is <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">bounded</a> by 16 <a href="/wiki/Cell_(mathematics)" class="mw-redirect" title="Cell (mathematics)">cells</a>, all of which are regular <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedra</a>.<a href="#cite_note-10">[e]</a> It has 32 <a href="/wiki/Triangle_(geometry)" class="mw-redirect" title="Triangle (geometry)">triangular</a> <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a>, 24 <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>, and 8 <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>. The 24 edges bound 6 <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> central squares lying on <a href="/wiki/Great_circles" class="mw-redirect" title="Great circles">great circles</a> in the 6 coordinate planes (3 pairs of completely orthogonal<a href="#cite_note-completely_orthogonal_planes-11">[f]</a> great squares). At each vertex, 3 great squares cross perpendicularly. The 6 edges meet at the vertex the way 6 edges meet at the <a href="/wiki/Apex_(geometry)" title="Apex (geometry)">apex</a> of a canonical <a href="/wiki/Octahedral_pyramid" title="Octahedral pyramid">octahedral pyramid</a>.<a href="#cite_note-octahedral_pyramid-9">[d]</a> The 6 orthogonal central planes of the 16-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> with 3 orthogonal great squares. <div class="mw-heading mw-heading3"><h3 id="Rotations">Rotations</h3>[<a href="/w/index.php?title=16-cell&action=edit&section=4" title="Edit section: Rotations">edit</a>]</div> <table class="wikitable" width="480"> <tbody><tr align="center" valign="top"> <td rowspan="2"><a href="/wiki/File:16-cell.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a0/16-cell.gif" decoding="async" width="255" height="255" class="mw-file-element" data-file-width="255" data-file-height="255" /></a> A 3D projection of a 16-cell performing a <a href="/wiki/SO(4)#Simple_rotations" class="mw-redirect" title="SO(4)">simple rotation</a> </td> <td><a href="/wiki/File:16-cell-orig.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/76/16-cell-orig.gif" decoding="async" width="256" height="256" class="mw-file-element" data-file-width="256" data-file-height="256" /></a> A 3D projection of a 16-cell performing a <a href="/wiki/SO(4)#Double_rotations" class="mw-redirect" title="SO(4)">double rotation</a> </td></tr></tbody></table> <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space" title="Rotations in 4-dimensional Euclidean space">Rotations in 4-dimensional Euclidean space</a> can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.<a href="#cite_note-FOOTNOTEKimRote20166§&nbsp;5._Four-Dimensional_Rotations-12">[6]</a> The 16-cell is a simple frame in which to observe 4-dimensional rotations, because each of the 16-cell's 6 great squares has another completely orthogonal great square (there are 3 pairs of completely orthogonal squares).<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> Many rotations of the 16-cell can be characterized by the angle of rotation in one of its great square planes (e.g. the xy plane) and another angle of rotation in the completely orthogonal great square plane (the wz plane).<a href="#cite_note-vertex_and_central_octahedra-16">[j]</a> Completely orthogonal great squares have disjoint vertices: 4 of the 16-cell's 8 vertices rotate in one plane, and the other 4 rotate independently in the completely orthogonal plane.<a href="#cite_note-Clifford_parallel_great_squares-13">[g]</a> In 2 or 3 dimensions a rotation is characterized by a single plane of rotation; this kind of rotation taking place in 4-space is called a <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space#Simple_rotations" title="Rotations in 4-dimensional Euclidean space">simple rotation</a>, in which only one of the two completely orthogonal planes rotates (the angle of rotation in the other plane is 0). In the 16-cell, a simple rotation in one of the 6 orthogonal planes moves only 4 of the 8 vertices; the other 4 remain fixed. (In the simple rotation animation above, all 8 vertices move because the plane of rotation is not one of the 6 orthogonal basis planes.) In a <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space#Double_rotations" title="Rotations in 4-dimensional Euclidean space">double rotation</a> both sets of 4 vertices move, but independently: the angles of rotation may be different in the 2 completely orthogonal planes. If the two angles happen to be the same, a maximally symmetric <a href="/wiki/Rotations_in_4-dimensional_Euclidean_space#Isoclinic_rotations" title="Rotations in 4-dimensional Euclidean space">isoclinic rotation</a> takes place.<a href="#cite_note-isoclinic_rotation-25">[q]</a> In the 16-cell an isoclinic rotation by 90 degrees of any pair of completely orthogonal square planes takes every square plane to its completely orthogonal square plane.<a href="#cite_note-exchange_of_completely_orthogonal_planes-27">[r]</a> <div class="mw-heading mw-heading3"><h3 id="Constructions">Constructions</h3>[<a href="/w/index.php?title=16-cell&action=edit&section=5" title="Edit section: Constructions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Octahedral_dipyramid">Octahedral dipyramid</h4>[<a href="/w/index.php?title=16-cell&action=edit&section=6" title="Edit section: Octahedral dipyramid">edit</a>]</div> <table class="wikitable floatright"> <tbody><tr> <th>Octahedron <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1001bcd6968314a965726f5dd193d1b11ada59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{3}}"> </th> <th>16-cell <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16fb7d2f7d9b9f310820b4e110b084003aa80fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{4}}"> </th></tr> <tr> <td><a href="/wiki/File:3-cube_t2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/3-cube_t2.svg/160px-3-cube_t2.svg.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/3-cube_t2.svg/240px-3-cube_t2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/3-cube_t2.svg/320px-3-cube_t2.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:4-demicube_t0_D4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/4-demicube_t0_D4.svg/160px-4-demicube_t0_D4.svg.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/4-demicube_t0_D4.svg/240px-4-demicube_t0_D4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/4-demicube_t0_D4.svg/320px-4-demicube_t0_D4.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td></tr> <tr> <td colspan="2">Orthogonal projections to skew hexagon hyperplane </td></tr></tbody></table> The simplest construction of the 16-cell is on the 3-dimensional cross polytope, the <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>. The octahedron has 3 perpendicular axes and 6 vertices in 3 opposite pairs (its <a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygon</a> is the <a href="/wiki/Hexagon" title="Hexagon">hexagon</a>). Add another pair of vertices, on a fourth axis perpendicular to all 3 of the other axes. Connect each new vertex to all 6 of the original vertices, adding 12 new edges. This raises two <a href="/wiki/Octahedral_pyramid" title="Octahedral pyramid">octahedral pyramids</a> on a shared octahedron base that lies in the 16-cell's central hyperplane.B_28-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973121§&nbsp;7.21._See_illustration_Fig_7.2B-28">[10]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Stereographic_polytope_16cell_colour.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stereographic_polytope_16cell_colour.png/220px-Stereographic_polytope_16cell_colour.png" decoding="async" width="220" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stereographic_polytope_16cell_colour.png/330px-Stereographic_polytope_16cell_colour.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stereographic_polytope_16cell_colour.png/440px-Stereographic_polytope_16cell_colour.png 2x" data-file-width="1100" data-file-height="1000" /></a><figcaption><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> of the 16-cell's 6 orthogonal central squares onto their great circles. Each circle is divided into 4 arc-edges at the intersections where 3 circles cross perpendicularly. Notice that each circle has one Clifford parallel circle that it does not intersect. Those two circles pass through each other like adjacent links in a chain.</figcaption></figure>The octahedron that the construction starts with has three perpendicular intersecting squares (which appear as rectangles in the hexagonal projections). Each square intersects with each of the other squares at two opposite vertices, with two of the squares crossing at each vertex. Then two more points are added in the fourth dimension (above and below the 3-dimensional hyperplane). These new vertices are connected to all the octahedron's vertices, creating 12 new edges and three more squares (which appear edge-on as the 3 diameters of the hexagon in the projection), and three more octahedra.<a href="#cite_note-octahedral_hyperplanes-14">[h]</a> Something unprecedented has also been created. Notice that each square no longer intersects with all of the other squares: it does intersect with four of them (with three of the squares crossing at each vertex now), but each square has one other square with which it shares no vertices: it is not directly connected to that square at all. These two separate perpendicular squares (there are three pairs of them) are like the opposite edges of a <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>: perpendicular, but non-intersecting. They lie opposite each other (parallel in some sense), and they don't touch, but they also pass through each other like two perpendicular links in a chain (but unlike links in a chain they have a common center). They are an example of Clifford parallel planes, and the 16-cell is the simplest regular polytope in which they occur. <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">Clifford</a> parallelism<a href="#cite_note-Clifford_parallels-20">[l]</a> of objects of more than one dimension (more than just curved lines) emerges here and occurs in all the subsequent 4-dimensional regular polytopes, where it can be seen as the defining relationship among disjoint concentric regular 4-polytopes and their corresponding parts. It can occur between congruent (similar) polytopes of 2 or more dimensions.<a href="#cite_note-FOOTNOTETyrrellSemple1971-29">[11]</a> For example, as noted <a href="#Geometry">above</a> all the subsequent convex regular 4-polytopes are compounds of multiple 16-cells; those 16-cells are <a href="/wiki/24-cell#Clifford_parallel_polytopes" title="24-cell">Clifford parallel polytopes</a>. <div class="mw-heading mw-heading4"><h4 id="Tetrahedral_constructions">Tetrahedral constructions</h4>[<a href="/w/index.php?title=16-cell&action=edit&section=7" title="Edit section: Tetrahedral constructions">edit</a>]</div> <table class="wikitable" width="480"> <tbody><tr align="center" valign="top"> <td><a href="/wiki/File:16-cell_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/16-cell_net.png/180px-16-cell_net.png" decoding="async" width="180" height="265" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/16-cell_net.png/270px-16-cell_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/16-cell_net.png/360px-16-cell_net.png 2x" data-file-width="802" data-file-height="1181" /></a> </td> <td><a href="/wiki/File:16-cell_nets.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/16-cell_nets.png/180px-16-cell_nets.png" decoding="async" width="180" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/16-cell_nets.png/270px-16-cell_nets.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/16-cell_nets.png/360px-16-cell_nets.png 2x" data-file-width="688" data-file-height="996" /></a> </td></tr></tbody></table> The 16-cell has two <a href="/wiki/Wythoff_construction" title="Wythoff construction">Wythoff constructions</a> from regular tetrahedra, a regular form and alternated form, shown here as <a href="/wiki/Net_(polyhedron)" title="Net (polyhedron)">nets</a>, the second represented by tetrahedral cells of two alternating colors. The alternated form is a <a href="#Symmetry_constructions">lower symmetry construction</a> of the 16-cell called the <a href="/wiki/Demitesseract" class="mw-redirect" title="Demitesseract">demitesseract</a>. Wythoff's construction replicates the 16-cell's <a href="/wiki/5-cell#Orthoschemes" title="5-cell">characteristic 5-cell</a> in a <a href="/wiki/Kaleidoscope" title="Kaleidoscope">kaleidoscope</a> of mirrors. Every regular 4-polytope has its characteristic 4-orthoscheme, an <a href="/wiki/5-cell#Irregular_5-cells" title="5-cell">irregular 5-cell</a>.<a href="#cite_note-characteristic_orthoscheme-30">[s]</a> There are three regular 4-polytopes with tetrahedral cells: the <a href="/wiki/5-cell" title="5-cell">5-cell</a>, the 16-cell, and the <a href="/wiki/600-cell" title="600-cell">600-cell</a>. Although all are bounded by regular tetrahedron cells, their characteristic 5-cells (4-orthoschemes) are different <a href="/wiki/5-cell#Isometries" title="5-cell">tetrahedral pyramids</a>, all based on the same characteristic irregular tetrahedron. They share the same <a href="/wiki/Tetrahedron#Orthoschemes" title="Tetrahedron">characteristic tetrahedron</a> (3-orthoscheme) and characteristic <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> (2-orthoscheme) because they have the same kind of cell.<a href="#cite_note-32">[t]</a> <table class="wikitable floatright"> <tbody><tr> <th colspan="6">Characteristics of the 16-cell4"_33-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973292–293Table_I(ii);_"16-cell,_𝛽4"-33">[13]</a> </th></tr> <tr> <th align="right"> </th> <th align="center">edge<a href="#cite_note-FOOTNOTECoxeter1973139§&nbsp;7.9_The_characteristic_simplex-34">[14]</a> </th> <th colspan="2" align="center">arc </th> <th colspan="2" align="center">dihedral<a href="#cite_note-FOOTNOTECoxeter1973290Table_I(ii);_"dihedral_angles"-35">[15]</a> </th></tr> <tr> <th align="right">𝒍 </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\approx 1.414}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>1.414</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\approx 1.414}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fafaf1870e0527a631858dcc0c7579b34ff297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.493ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\approx 1.414}"> </td> <td align="center">90° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e31a202557dfbf326b44ebcc914ba3ab08fff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}"> </td> <td align="center">120° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d347f5b40db81e556675c27ed6da65079059c5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2\pi }{3}}}"> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th align="right">𝟀 </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {2}{3}}}\approx 0.816}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.816</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {2}{3}}}\approx 0.816}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf13a57252e143fcdd2e2c351fe282bb17951d05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {2}{3}}}\approx 0.816}"> </td> <td align="center">60″ </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e0953cd4cd6fa5a3f4b473c8bd0cf32bbb5c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{3}}}"> </td> <td align="center">60° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e0953cd4cd6fa5a3f4b473c8bd0cf32bbb5c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{3}}}"> </td></tr> <tr> <th align="right">𝝉<a href="#cite_note-reversed_greek_symbols-36">[u]</a> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.707</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4dc0f6527d5ac3f56ed3e68bc66e0b2f6a2bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> </td> <td align="center">45″ </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f065c03576345b1b536c3cd513af53e8c35c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{4}}}"> </td> <td align="center">45° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f065c03576345b1b536c3cd513af53e8c35c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{4}}}"> </td></tr> <tr> <th align="right">𝟁 </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.408</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfffe1715d3520123a306ce41d30c1506cdb359c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}"> </td> <td align="center">30″ </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75bc23c5dc0aac6e2066ae76ee4a5cf52fd3e19b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{6}}}"> </td> <td align="center">60° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e0953cd4cd6fa5a3f4b473c8bd0cf32bbb5c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{3}}}"> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{0}R^{3}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{0}R^{3}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6e3551246aea0817396c3a3d4d7bde3e48ee1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{0}R^{3}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {3}{4}}}\approx 0.866}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.866</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {3}{4}}}\approx 0.866}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a005250a320af081f18edf9f13f703e7e1ae2d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {3}{4}}}\approx 0.866}"> </td> <td align="center">60° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e0953cd4cd6fa5a3f4b473c8bd0cf32bbb5c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{3}}}"> </td> <td align="center">90° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e31a202557dfbf326b44ebcc914ba3ab08fff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}"> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{1}R^{3}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{1}R^{3}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752166c1b2cdcd24b56f0adae04c08a5e75095e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{1}R^{3}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5155145fa08cfb3a1775408bc256aed546249a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.052ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}"> </td> <td align="center">45° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f065c03576345b1b536c3cd513af53e8c35c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{4}}}"> </td> <td align="center">90° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e31a202557dfbf326b44ebcc914ba3ab08fff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}"> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{2}R^{3}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{2}R^{3}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1741983f4e62c25f7756064b6d3ec9f23e07174d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{2}R^{3}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{12}}}\approx 0.289}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.289</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{12}}}\approx 0.289}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab247b688ce48ab225ab0bc7c7e3d34f5ff121c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.199ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{12}}}\approx 0.289}"> </td> <td align="center">30° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75bc23c5dc0aac6e2066ae76ee4a5cf52fd3e19b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{6}}}"> </td> <td align="center">90° </td> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e31a202557dfbf326b44ebcc914ba3ab08fff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}"> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{0}R^{4}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{0}R^{4}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6aa7aaf454b849082b05af6fc128463286d6f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{0}R^{4}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{1}R^{4}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{1}R^{4}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a02707dfe259850f534490492d56be62d9d6a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{1}R^{4}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.707</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4dc0f6527d5ac3f56ed3e68bc66e0b2f6a2bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{2}R^{4}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{2}R^{4}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af1bfc7856b5a75361749084f8066a943bb83c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{2}R^{4}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>≈</mo> <mn>0.577</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f44d4356ecdfc00e412b5335ed4d4aa185d1d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td></tr> <tr> <th align="right"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{3}R^{4}/l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{3}R^{4}/l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54facf048252ac15276ae6927b6547339c6e0920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.728ex; height:3.176ex;" alt="{\displaystyle _{3}R^{4}/l}"> </th> <td align="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5155145fa08cfb3a1775408bc256aed546249a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.052ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{4}}}=0.5}"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td></tr></tbody></table> The characteristic 5-cell of the regular 16-cell is represented by the <a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter-Dynkin diagram</a> <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" /><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" /><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" /><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" /><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" /><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" /><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" />, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular <a href="/wiki/Pyramid_(mathematics)#Polyhedral_pyramid" class="mw-redirect" title="Pyramid (mathematics)">tetrahedral pyramid</a> based on the <a href="/wiki/Tetrahedron#Orthoschemes" title="Tetrahedron">characteristic tetrahedron of the regular tetrahedron</a>. The regular 16-cell is subdivided by its symmetry hyperplanes into 384 instances of its characteristic 5-cell that all meet at its center. The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 16-cell).<a href="#cite_note-characteristic_radii-37">[v]</a> If the regular 16-cell has unit radius edge and edge length 𝒍 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" 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="{\displaystyle {\sqrt {2}}}">, its characteristic 5-cell's ten edges have lengths <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {2}{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {2}{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789f1d9736aaedeb05b88d1125c9a92b9aab8a59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {2}{3}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bf07a2c9eeeaff20adb64ac6938e0811092816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{6}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{6}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bee5226c00a4711eb77e88c85ef1724948334be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{6}}}}"> around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),<a href="#cite_note-reversed_greek_symbols-36">[u]</a> plus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {3}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {3}{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f323af25fb6f7a21aa8bd372598addb848c896ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {3}{4}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dfb83a9805c355478a59695613487054137507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{4}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{12}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{12}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c691be99eb817fb48755cef319cb30f3a828ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.804ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{12}}}}"> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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="{\displaystyle 1}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bf07a2c9eeeaff20adb64ac6938e0811092816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f7365d50f1f5a445e3b3915cf4b9823f5f3f1ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{3}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dfb83a9805c355478a59695613487054137507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{4}}}}"> (edges which are the characteristic radii of the regular 16-cell). The 4-edge path along orthogonal edges of the orthoscheme is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97bf07a2c9eeeaff20adb64ac6938e0811092816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{2}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{6}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{6}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bee5226c00a4711eb77e88c85ef1724948334be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{6}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dfb83a9805c355478a59695613487054137507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{4}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\tfrac {1}{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\tfrac {1}{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dfb83a9805c355478a59695613487054137507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.982ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\tfrac {1}{4}}}}">, first from a 16-cell vertex to a 16-cell edge center, then turning 90° to a 16-cell face center, then turning 90° to a 16-cell tetrahedral cell center, then turning 90° to the 16-cell center. <div class="mw-heading mw-heading4"><h4 id="Helical_construction">Helical construction</h4>[<a href="/w/index.php?title=16-cell&action=edit&section=8" title="Edit section: Helical construction">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eight_face-bonded_tetrahedra.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Eight_face-bonded_tetrahedra.jpg/220px-Eight_face-bonded_tetrahedra.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Eight_face-bonded_tetrahedra.jpg/330px-Eight_face-bonded_tetrahedra.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Eight_face-bonded_tetrahedra.jpg/440px-Eight_face-bonded_tetrahedra.jpg 2x" data-file-width="640" data-file-height="480" /></a><figcaption>A 4-dimensional ring of 8 face-bonded tetrahedra, seen in the <a href="/wiki/Boerdijk%E2%80%93Coxeter_helix" title="Boerdijk–Coxeter helix">Boerdijk–Coxeter helix</a>, bounded by three eight-edge circular paths of different colors, cut and laid out flat in 3-dimensional space. It contains an isocline axis (not shown), a helical circle of circumference 4𝝅 that twists through all four dimensions and visits all 8 vertices.<a href="#cite_note-isocline-23">[o]</a> The two blue-blue-yellow triangles at either end of the cut ring are the same object.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:16-cell_8-ring_net4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/16-cell_8-ring_net4.png/220px-16-cell_8-ring_net4.png" decoding="async" width="220" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/16-cell_8-ring_net4.png/330px-16-cell_8-ring_net4.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/16-cell_8-ring_net4.png/440px-16-cell_8-ring_net4.png 2x" data-file-width="1514" data-file-height="838" /></a><figcaption>Net and orthogonal projection</figcaption></figure> A 16-cell can be constructed (three different ways) from two <a href="/wiki/Boerdijk%E2%80%93Coxeter_helix" title="Boerdijk–Coxeter helix">Boerdijk–Coxeter helixes</a> of eight chained tetrahedra, each bent in the fourth dimension into a ring.<a href="#cite_note-FOOTNOTECoxeter197045Table_2:_Reflexible_honeycombs_and_their_groups-38">[16]</a><a href="#cite_note-FOOTNOTEBanchoff2013-39">[17]</a> The two circular helixes spiral around each other, nest into each other and pass through each other forming a <a href="/wiki/Hopf_link" title="Hopf link">Hopf link</a>. The 16 triangle faces can be seen in a 2D net within a <a href="/wiki/Triangular_tiling" title="Triangular tiling">triangular tiling</a>, with 6 triangles around every vertex. The purple edges represent the <a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygon</a> of the 16-cell. The eight-cell ring of tetrahedra contains three <a href="/wiki/Octagram" title="Octagram">octagrams</a> of different colors, eight-edge circular paths that wind twice around the 16-cell on every third vertex of the octagram. The orange and yellow edges are two four-edge halves of one octagram, which join their ends to form a <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a>. Thus the 16-cell can be decomposed into two cell-disjoint circular chains of eight tetrahedrons each, four edges long, one spiraling to the right (clockwise) and the other spiraling to the left (counterclockwise). The left-handed and right-handed cell rings fit together, nesting into each other and entirely filling the 16-cell, even though they are of opposite chirality. This decomposition can be seen in a 4-4 <a href="/wiki/Duoantiprism" class="mw-redirect" title="Duoantiprism">duoantiprism</a> construction of the 16-cell: <img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> or <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" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" />, <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {2}⨂{2} or s{2}s{2}, <a href="/wiki/Coxeter_notation" title="Coxeter notation">symmetry</a> [4,2+,4], order 64. Three eight-edge paths (of different colors) spiral along each eight-cell ring, making 90° angles at each vertex. (In the Boerdijk–Coxeter helix before it is bent into a ring, the angles in different paths vary, but are not 90°.) Three paths (with three different colors and apparent angles) pass through each vertex. When the helix is bent into a ring, the segments of each eight-edge path (of various lengths) join their ends, forming a Möbius strip eight edges long along its single-sided circumference of 4𝝅, and one edge wide.<a href="#cite_note-Möbius_circle-24">[p]</a> The six four-edge halves of the three eight-edge paths each make four 90° angles, but they are not the six orthogonal great squares: they are open-ended squares, four-edge 360° helices whose open ends are <a href="/wiki/Antipodal_point" title="Antipodal point">antipodal</a> vertices. The four edges come from four different great squares, and are mutually orthogonal. Combined end-to-end in pairs of the same <a href="/wiki/Chirality" title="Chirality">chirality</a>, the six four-edge paths make three eight-edge Möbius loops, <a href="/wiki/Helix" title="Helix">helical</a> octagrams. Each octagram is both a <a href="/wiki/Petrie_polygon" title="Petrie polygon">Petrie polygon</a> of the 16-cell, and the helical track along which all eight vertices rotate together, in one of the 16-cell's distinct isoclinic <a href="#Rotations">rotations</a>.<a href="#cite_note-only_one_disjoint_pair_of_eight-cell_rings-40">[w]</a> <table class="wikitable" width="610"> <tbody><tr> <th colspan="5">Five ways of looking at the same <a href="/wiki/Skew_polygon" title="Skew polygon">skew</a> <a href="/wiki/Octagram" title="Octagram">octagram</a><a href="#cite_note-41">[x]</a> </th></tr> <tr> <th><a href="#Rotations">Edge path</a> </th> <th><a href="/wiki/Petrie_polygon#The_Petrie_polygon_of_regular_polychora_(4-polytopes)" title="Petrie polygon">Petrie polygon</a>1''_42-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973292–293Table_I(ii);_24-cell_''h1''-42">[18]</a> </th> <th>16-cell </th> <th><a href="/wiki/Hopf_fibration" title="Hopf fibration">Discrete fibration</a> </th> <th><a href="#Coordinates">Diameter chords</a> </th></tr> <tr> <th><a href="/wiki/Octagram" title="Octagram">Octagram</a>{8/3}2''_43-0" class="reference"><a href="#cite_note-FOOTNOTECoxeter1973292–293Table_I(ii);_24-cell_''h2''-43">[19]</a> </th> <th><a href="/wiki/Petrie_polygon#The_Petrie_polygon_of_regular_polychora_(4-polytopes)" title="Petrie polygon">Octagram</a>{8/1} </th> <th><a href="/wiki/Coxeter_element#Coxeter_plane" title="Coxeter element">Coxeter plane</a> <a href="/wiki/B4_polytope" title="B4 polytope">B4</a> </th> <th><a href="/wiki/Octagram#Star_polygon_compounds" title="Octagram">Octagram</a>{8/2}=2{4} </th> <th><a href="/wiki/Octagram#Star_polygon_compounds" title="Octagram">Octagram</a>{8/4}=4{2} </th></tr> <tr> <td align="center"><a href="/wiki/File:16-cell_skew_octagram_(8-3).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/16-cell_skew_octagram_%288-3%29.png/120px-16-cell_skew_octagram_%288-3%29.png" decoding="async" width="120" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/16-cell_skew_octagram_%288-3%29.png/180px-16-cell_skew_octagram_%288-3%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/16-cell_skew_octagram_%288-3%29.png/240px-16-cell_skew_octagram_%288-3%29.png 2x" data-file-width="1373" data-file-height="1386" /></a> </td> <td align="center"><a href="/wiki/File:16-cell_skew_octagram_(8).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/16-cell_skew_octagram_%288%29.png/120px-16-cell_skew_octagram_%288%29.png" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/16-cell_skew_octagram_%288%29.png/180px-16-cell_skew_octagram_%288%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/16-cell_skew_octagram_%288%29.png/240px-16-cell_skew_octagram_%288%29.png 2x" data-file-width="1367" data-file-height="1354" /></a> </td> <td align="center"><a href="/wiki/File:16-cell_skew_octagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/16-cell_skew_octagram.png/120px-16-cell_skew_octagram.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/16-cell_skew_octagram.png/180px-16-cell_skew_octagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/16-cell_skew_octagram.png/240px-16-cell_skew_octagram.png 2x" data-file-width="1354" data-file-height="1357" /></a> </td> <td align="center"><a href="/wiki/File:16-cell_skew_octagram_2(4).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/16-cell_skew_octagram_2%284%29.png/120px-16-cell_skew_octagram_2%284%29.png" decoding="async" width="120" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/16-cell_skew_octagram_2%284%29.png/180px-16-cell_skew_octagram_2%284%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/16-cell_skew_octagram_2%284%29.png/240px-16-cell_skew_octagram_2%284%29.png 2x" data-file-width="1353" data-file-height="1380" /></a> </td> <td align="center"><a href="/wiki/File:16-cell_skew_octagram_4(2).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/16-cell_skew_octagram_4%282%29.png/120px-16-cell_skew_octagram_4%282%29.png" decoding="async" width="120" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/16-cell_skew_octagram_4%282%29.png/180px-16-cell_skew_octagram_4%282%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/16-cell_skew_octagram_4%282%29.png/240px-16-cell_skew_octagram_4%282%29.png 2x" data-file-width="1358" data-file-height="1392" /></a> </td></tr> <tr> <td>The eight √2 chords of the edge-path of an isocline.<a href="#cite_note-isocline_curve-44">[y]</a> </td> <td>Skew <a href="/wiki/Octagon" title="Octagon">octagon</a> of eight √2 edges. The 16-cell has 3 of these 8-vertex circuits. </td> <td>All 24 √2 edges and the four √4 orthogonal axes. </td> <td>Two completely orthogonal (disjoint) great squares of √2 edges.<a href="#cite_note-Clifford_parallel_great_squares-13">[g]</a> </td> <td>The four √4 chords of an isocline. Every fourth isocline vertex is joined to its antipodal vertex by a 16-cell axis.<a href="#cite_note-isocline_curve-44">[y]</a> </td></tr></tbody></table> Each eight-edge helix is a <a href="/wiki/Skew_polygon" title="Skew polygon">skew</a> <a href="/wiki/Octagram" title="Octagram">octagram</a>{8/3} that <a href="/wiki/Winding_number" title="Winding number">winds three times</a> around the 16-cell and visits every vertex before closing into a loop. Its eight √2 edges are chords of an isocline, a helical arc on which the 8 vertices circle during an isoclinic rotation.<a href="#cite_note-Möbius_circle-24">[p]</a> All eight 16-cell vertices are √2 apart except for opposite (antipodal) vertices, which are √4 apart. A vertex moving on the isocline visits three other vertices that are √2 apart before reaching the fourth vertex that is √4 away.<a href="#cite_note-isocline-23">[o]</a> The eight-cell ring is <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a>: there is a right-handed form which spirals clockwise, and a left-handed form which spirals counterclockwise. The 16-cell contains one of each, so it also contains a left and a right isocline; the isocline is the circular axis around which the eight-cell ring twists. Each isocline visits all eight vertices of the 16-cell.<a href="#cite_note-each_16-cell_isocline_reaches_all_8_vertices-47">[ab]</a> Each eight-cell ring contains half of the 16 cells, but all 8 vertices; the two rings share the vertices, as they nest into each other and fit together. They also share the 24 edges, though left and right octagram helices are different eight-edge paths.<a href="#cite_note-48">[ac]</a> Because there are three pairs of completely orthogonal great squares,<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> there are three congruent ways to compose a 16-cell from two eight-cell rings. The 16-cell contains three left-right pairs of eight-cell rings in different orientations, with each cell ring containing its axial isocline.<a href="#cite_note-only_one_disjoint_pair_of_eight-cell_rings-40">[w]</a> Each left-right pair of isoclines is the track of a left-right pair of distinct isoclinic rotations: the rotations in one pair of completely orthogonal invariant planes of rotation.<a href="#cite_note-Clifford_parallel_great_squares-13">[g]</a> At each vertex, there are three great squares and six octagram isoclines that cross at the vertex and share a 16-cell axis chord.<a href="#cite_note-50">[ad]</a> <div class="mw-heading mw-heading3"><h3 id="As_a_configuration">As a configuration</h3>[<a href="/w/index.php?title=16-cell&action=edit&section=9" title="Edit section: As a configuration">edit</a>]</div> This <a href="/wiki/Regular_4-polytope#As_configurations" title="Regular 4-polytope">configuration matrix</a> represents the 16-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 16-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>24</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>16</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cc51bad7b455e513bb3ca683737241c9943ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:19.709ex; height:12.509ex;" alt="{\displaystyle {\begin{bmatrix}{\begin{matrix}8&6&12&8\\2&24&4&4\\3&3&32&2\\4&6&4&16\end{matrix}}\end{bmatrix}}}"> <div class="mw-heading mw-heading2"><h2 id="Tessellations">Tessellations</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=10" title="Edit section: Tessellations">edit</a>]</div> One can <a href="/wiki/Tessellation" title="Tessellation">tessellate</a> 4-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> by regular 16-cells. This is called the <a href="/wiki/16-cell_honeycomb" title="16-cell honeycomb">16-cell honeycomb</a> and has <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> {3,3,4,3}. Hence, the 16-cell has a <a href="/wiki/Dihedral_angle" title="Dihedral angle">dihedral angle</a> of 120°.<a href="#cite_note-FOOTNOTECoxeter1973293-51">[21]</a> Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point. Twenty-four 16-cells meet at any given vertex in this tessellation. The dual tessellation, the <a href="/wiki/24-cell_honeycomb" title="24-cell honeycomb">24-cell honeycomb</a>, {3,4,3,3}, is made of regular <a href="/wiki/24-cell" title="24-cell">24-cells</a>. Together with the <a href="/wiki/Tesseractic_honeycomb" title="Tesseractic honeycomb">tesseractic honeycomb</a> {4,3,3,4} these are the only three <a href="/wiki/List_of_regular_polytopes#Tessellations_of_Euclidean_4-space" title="List of regular polytopes">regular tessellations</a> of R4. <div class="mw-heading mw-heading2"><h2 id="Projections">Projections</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=11" title="Edit section: Projections">edit</a>]</div> <table class="wikitable"> <caption><a href="/wiki/Orthographic_projection" title="Orthographic projection">orthographic projections</a> </caption> <tbody><tr style="text-align:center;"> <th><a href="/wiki/Coxeter_plane" class="mw-redirect" title="Coxeter plane">Coxeter plane</a> </th> <th>B4 </th> <th>B3 / D4 / A2 </th> <th>B2 / D3 </th></tr> <tr style="text-align:center;"> <th>Graph </th> <td><a href="/wiki/File:4-cube_t3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/150px-4-cube_t3.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/225px-4-cube_t3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/300px-4-cube_t3.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:4-demicube_t0_D4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/4-demicube_t0_D4.svg/150px-4-demicube_t0_D4.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/4-demicube_t0_D4.svg/225px-4-demicube_t0_D4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/4-demicube_t0_D4.svg/300px-4-demicube_t0_D4.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:4-cube_t3_B2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/4-cube_t3_B2.svg/150px-4-cube_t3_B2.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/4-cube_t3_B2.svg/225px-4-cube_t3_B2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/4-cube_t3_B2.svg/300px-4-cube_t3_B2.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td></tr> <tr style="text-align:center;"> <th><a href="/wiki/Dihedral_symmetry" class="mw-redirect" title="Dihedral symmetry">Dihedral symmetry</a> </th> <td>[8] </td> <td>[6] </td> <td>[4] </td></tr> <tr style="text-align:center;"> <th>Coxeter plane </th> <th>F4 </th> <th>A3 </th></tr> <tr style="text-align:center;"> <th>Graph </th> <td><a href="/wiki/File:4-cube_t3_F4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/4-cube_t3_F4.svg/150px-4-cube_t3_F4.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/4-cube_t3_F4.svg/225px-4-cube_t3_F4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/4-cube_t3_F4.svg/300px-4-cube_t3_F4.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td> <td><a href="/wiki/File:4-cube_t3_A3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/4-cube_t3_A3.svg/150px-4-cube_t3_A3.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/4-cube_t3_A3.svg/225px-4-cube_t3_A3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/4-cube_t3_A3.svg/300px-4-cube_t3_A3.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a> </td></tr> <tr style="text-align:center;"> <th>Dihedral symmetry </th> <td>[12/3] </td> <td>[4] </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Orthogonal_projection_envelopes_16-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Orthogonal_projection_envelopes_16-cell.png/220px-Orthogonal_projection_envelopes_16-cell.png" decoding="async" width="220" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Orthogonal_projection_envelopes_16-cell.png/330px-Orthogonal_projection_envelopes_16-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Orthogonal_projection_envelopes_16-cell.png/440px-Orthogonal_projection_envelopes_16-cell.png 2x" data-file-width="1580" data-file-height="1915" /></a><figcaption>Projection envelopes of the 16-cell. (Each cell is drawn with different color faces, inverted cells are undrawn)</figcaption></figure> The cell-first parallel projection of the 16-cell into 3-space has a <a href="/wiki/Cube" title="Cube">cubical</a> envelope. The closest and farthest cells are projected to inscribed tetrahedra within the cube, corresponding with the two possible ways to inscribe a regular tetrahedron in a cube. Surrounding each of these tetrahedra are 4 other (non-regular) tetrahedral volumes that are the images of the 4 surrounding tetrahedral cells, filling up the space between the inscribed tetrahedron and the cube. The remaining 6 cells are projected onto the square faces of the cube. In this projection of the 16-cell, all its edges lie on the faces of the cubical envelope. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cell-first_parallel_projection_of_16-cell.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Cell-first_parallel_projection_of_16-cell.gif/220px-Cell-first_parallel_projection_of_16-cell.gif" decoding="async" width="220" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Cell-first_parallel_projection_of_16-cell.gif/330px-Cell-first_parallel_projection_of_16-cell.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Cell-first_parallel_projection_of_16-cell.gif/440px-Cell-first_parallel_projection_of_16-cell.gif 2x" data-file-width="720" data-file-height="782" /></a><figcaption>Cell-first parallel projection of sections of 16-cell</figcaption></figure> The cell-first perspective projection of the 16-cell into 3-space has a <a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedral</a> envelope. The layout of the cells within this envelope are analogous to that of the cell-first parallel projection. The vertex-first parallel <a href="/wiki/Graphical_projection" class="mw-redirect" title="Graphical projection">projection</a> of the 16-cell into 3-space has an <a href="/wiki/Octahedron" title="Octahedron">octahedral</a> <a href="/wiki/Projection_envelope" class="mw-redirect" title="Projection envelope">envelope</a>. This octahedron can be divided into 8 tetrahedral volumes, by cutting along the coordinate planes. Each of these volumes is the image of a pair of cells in the 16-cell. The closest vertex of the 16-cell to the viewer projects onto the center of the octahedron. Finally the edge-first parallel projection has a shortened octahedral envelope, and the face-first parallel projection has a <a href="/wiki/Hexagonal_bipyramid" title="Hexagonal bipyramid">hexagonal bipyramidal</a> envelope. <div class="mw-heading mw-heading2"><h2 id="4_sphere_Venn_diagram">4 sphere Venn diagram</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=12" title="Edit section: 4 sphere Venn diagram">edit</a>]</div> A 3-dimensional projection of the 16-cell and 4 intersecting spheres (a <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> of 4 sets) are <a href="/wiki/Topology" title="Topology">topologically</a> equivalent. <table> <tbody><tr> <td> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti 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class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_weight_2%2C_solid.png/134px-4_spheres%2C_weight_2%2C_solid.png" decoding="async" width="134" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_weight_2%2C_solid.png/201px-4_spheres%2C_weight_2%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/4_spheres%2C_weight_2%2C_solid.png/268px-4_spheres%2C_weight_2%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></div></div><div class="tsingle" style="width:136px;max-width:136px"><div class="thumbimage" style="height:145px;overflow:hidden"><a href="/wiki/File:4_spheres,_weight_3,_solid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/4_spheres%2C_weight_3%2C_solid.png/134px-4_spheres%2C_weight_3%2C_solid.png" decoding="async" width="134" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/4_spheres%2C_weight_3%2C_solid.png/201px-4_spheres%2C_weight_3%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/4_spheres%2C_weight_3%2C_solid.png/268px-4_spheres%2C_weight_3%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></div></div><div class="tsingle" style="width:136px;max-width:136px"><div class="thumbimage" style="height:145px;overflow:hidden"><a href="/wiki/File:4_spheres,_cell_15,_solid.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4_spheres%2C_cell_15%2C_solid.png/134px-4_spheres%2C_cell_15%2C_solid.png" decoding="async" width="134" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4_spheres%2C_cell_15%2C_solid.png/201px-4_spheres%2C_cell_15%2C_solid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4_spheres%2C_cell_15%2C_solid.png/268px-4_spheres%2C_cell_15%2C_solid.png 2x" data-file-width="4000" data-file-height="4320" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The 16 cells ordered by number of intersecting spheres (from 0 to 4)     (see all <a href="https://commons.wikimedia.org/wiki/Category:Venn_diagrams_rgby;_single_cells" class="extiw" title="commons:Category:Venn diagrams rgby; single cells">cells</a> and <a href="https://en.wikiversity.org/wiki/Tesseract_and_16-cell_faces" class="extiw" title="v:Tesseract and 16-cell faces">k-faces</a>)</div></div></div></div> </td> <td> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:282px;max-width:282px"><div class="trow"><div class="tsingle" style="width:131px;max-width:131px"><div class="thumbimage" style="height:131px;overflow:hidden"><a href="/wiki/File:4_spheres_as_rings,_vertical.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/4_spheres_as_rings%2C_vertical.png/129px-4_spheres_as_rings%2C_vertical.png" decoding="async" width="129" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/4_spheres_as_rings%2C_vertical.png/194px-4_spheres_as_rings%2C_vertical.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/4_spheres_as_rings%2C_vertical.png/258px-4_spheres_as_rings%2C_vertical.png 2x" data-file-width="4000" data-file-height="4080" /></a></div></div><div class="tsingle" style="width:147px;max-width:147px"><div class="thumbimage" style="height:131px;overflow:hidden"><a href="/wiki/File:Stereographic_polytope_16cell.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Stereographic_polytope_16cell.png/145px-Stereographic_polytope_16cell.png" decoding="async" width="145" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Stereographic_polytope_16cell.png/218px-Stereographic_polytope_16cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Stereographic_polytope_16cell.png/290px-Stereographic_polytope_16cell.png 2x" data-file-width="1100" data-file-height="1000" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">4 sphere Venn diagram and 16-cell projection in the same orientation</div></div></div></div> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Symmetry_constructions">Symmetry constructions</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=13" title="Edit section: Symmetry constructions">edit</a>]</div> The 16-cell's <a href="/wiki/Coxeter_group" title="Coxeter group">symmetry group</a> is denoted <a href="/wiki/B4_polytope" title="B4 polytope">B4</a>. There is a lower symmetry form of the 16-cell, called a demitesseract or 4-demicube, a member of the <a href="/wiki/Demihypercube" title="Demihypercube">demihypercube</a> family, and represented by h{4,3,3}, and <a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagrams</a> <img src="//upload.wikimedia.org/wikipedia/commons/4/48/CDel_node_h1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> or <img src="//upload.wikimedia.org/wikipedia/commons/f/fc/CDel_nodes_10ru.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><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" /><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" /><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" />. It can be drawn bicolored with alternating <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedral</a> cells. It can also be seen in lower symmetry form as a tetrahedral antiprism, constructed by 2 parallel <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedra</a> in dual configurations, connected by 8 (possibly elongated) tetrahedra. It is represented by s{2,4,3}, and Coxeter diagram: <img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" />. It can also be seen as a snub 4-<a href="/wiki/Orthotope" class="mw-redirect" title="Orthotope">orthotope</a>, represented by s{21,1,1}, and Coxeter diagram: <img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> or <img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/3/3c/CDel_split1-22.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_nodes_hh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" />. With the <a href="/wiki/Tesseract" title="Tesseract">tesseract</a> constructed as a 4-4 <a href="/wiki/Duoprism" title="Duoprism">duoprism</a>, the 16-cell can be seen as its dual, a 4-4 <a href="/wiki/Duopyramid" title="Duopyramid">duopyramid</a>. <table class="wikitable"> <tbody><tr> <th>Name </th> <th><a href="/wiki/Coxeter_diagram" class="mw-redirect" title="Coxeter diagram">Coxeter diagram</a> </th> <th><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <th><a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> </th> <th>Order </th> <th><a href="/wiki/Vertex_figure" title="Vertex figure">Vertex figure</a> </th></tr> <tr align="center"> <th>Regular 16-cell </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>{3,3,4} </td> <td>[3,3,4]</td> <td>384 </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /> </td></tr> <tr align="center"> <th>Demitesseract <a href="/wiki/Quasiregular_polytope" class="mw-redirect" title="Quasiregular polytope">Quasiregular</a> 16-cell </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/f/fc/CDel_nodes_10ru.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><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" /><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" /><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" /> = <img src="//upload.wikimedia.org/wikipedia/commons/4/48/CDel_node_h1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><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" /> <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> = <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/5/5d/CDel_node_h0.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td>h{4,3,3} {3,31,1} </td> <td>[31,1,1] = [1+,4,3,3]</td> <td>192 </td> <td><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" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /> </td></tr> <tr align="center"> <th>Alternated 4-4 <a href="/wiki/Duoprism" title="Duoprism">duoprism</a> </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/b/bc/CDel_label2.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/6/63/CDel_branch_hh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/7/7a/CDel_4a4b.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /> </td> <td>2s{4,2,4} </td> <td>[[4,2+,4]]</td> <td>64 </td> <td> </td></tr> <tr align="center"> <th>Tetrahedral antiprism </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /><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" /><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" /> </td> <td>s{2,4,3} </td> <td>[2+,4,3]</td> <td>48 </td> <td> </td></tr> <tr align="center"> <th>Alternated square prism prism </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><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" /> </td> <td>sr{2,2,4} </td> <td>[(2,2)+,4]</td> <td>16 </td> <td> </td></tr> <tr align="center"> <th>Snub 4-<a href="/wiki/Orthotope" class="mw-redirect" title="Orthotope">orthotope</a> </th> <td><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> = <img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/3/3c/CDel_split1-22.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /><img src="//upload.wikimedia.org/wikipedia/commons/9/9c/CDel_nodes_hh.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td> <td>s{21,1,1} </td> <td>[2,2,2]+ = [21,1,1]+</td> <td>8 </td> <td><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /> </td></tr> <tr align="center"> <th rowspan="6">4-<a href="/wiki/Rhombic_fusil" class="mw-redirect" title="Rhombic fusil">fusil</a> </th></tr> <tr align="center"> <td><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>{3,3,4} </td> <td>[3,3,4]</td> <td>384 </td> <td><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" /><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" /><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" /><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" /><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" /> </td></tr> <tr align="center"> <td><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>{4}+{4} or 2{4} </td> <td>[[4,2,4]] = [8,2+,8]</td> <td>128 </td> <td><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" /><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" /><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" /><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" /><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" /> </td></tr> <tr align="center"> <td><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>{3,4}+{ } </td> <td>[4,3,2]</td> <td>96 </td> <td><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" /><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" /><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" /><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" /><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" /> <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" /><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" /><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" /><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" /><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" /> </td></tr> <tr align="center"> <td><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>{4}+2{ } </td> <td>[4,2,2]</td> <td>32 </td> <td><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" /><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" /><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" /><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" /><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" /> <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" /><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" /><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" /><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" /><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" /> </td></tr> <tr align="center"> <td><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" /><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" /><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" /><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" /><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" /><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" /><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" /> </td> <td>{ }+{ }+{ }+{ } or 4{ } </td> <td>[2,2,2]</td> <td>16 </td> <td><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" /><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" /><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" /><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" /><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" /> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Related_complex_polygons">Related complex polygons</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=14" title="Edit section: Related complex polygons">edit</a>]</div> The <a href="/wiki/M%C3%B6bius%E2%80%93Kantor_polygon" title="Möbius–Kantor polygon">Möbius–Kantor polygon</a> is a <a href="/wiki/Regular_complex_polytope" class="mw-redirect" title="Regular complex polytope">regular complex polygon</a> 3{3}3, <img src="//upload.wikimedia.org/wikipedia/commons/5/57/CDel_3node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/4/4a/CDel_3node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" />, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}"> shares the same vertices as the 16-cell. It has 8 vertices, and 8 3-edges.<a href="#cite_note-FOOTNOTECoxeter199130,_47-52">[22]</a><a href="#cite_note-FOOTNOTECoxeterShephard1992-53">[23]</a> The regular complex polygon, 2{4}4, <img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /><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" /><img src="//upload.wikimedia.org/wikipedia/commons/0/07/CDel_4node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" />, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}"> has a real representation as a 16-cell in 4-dimensional space with 8 vertices, 16 2-edges, only half of the edges of the 16-cell. Its symmetry is 4[4]2, order 32.<a href="#cite_note-FOOTNOTECoxeter1991108-54">[24]</a> <table class="wikitable" width="320"> <caption><a href="/wiki/Orthographic_projection" title="Orthographic projection">Orthographic projections</a> of 2{4}4 polygon </caption> <tbody><tr valign="top"> <td><a href="/wiki/File:Complex_polygon_2-4-4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Complex_polygon_2-4-4.png/160px-Complex_polygon_2-4-4.png" decoding="async" width="160" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Complex_polygon_2-4-4.png/240px-Complex_polygon_2-4-4.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Complex_polygon_2-4-4.png/320px-Complex_polygon_2-4-4.png 2x" data-file-width="580" data-file-height="568" /></a> In B4 <a href="/wiki/Coxeter_plane" class="mw-redirect" title="Coxeter plane">Coxeter plane</a>, 2{4}4 has 8 vertices and 16 2-edges, shown here with 4 sets of colors. </td> <td><a href="/wiki/File:Complex_polygon_2-4-4_bipartite_graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Complex_polygon_2-4-4_bipartite_graph.png/160px-Complex_polygon_2-4-4_bipartite_graph.png" decoding="async" width="160" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Complex_polygon_2-4-4_bipartite_graph.png/240px-Complex_polygon_2-4-4_bipartite_graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Complex_polygon_2-4-4_bipartite_graph.png/320px-Complex_polygon_2-4-4_bipartite_graph.png 2x" data-file-width="580" data-file-height="568" /></a> The 8 vertices are grouped in 2 sets (shown red and blue), each only connected with edges to vertices in the other set, making this polygon a <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite graph</a>, K4,4.<a href="#cite_note-FOOTNOTECoxeter1991114-55">[25]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Related_uniform_polytopes_and_honeycombs">Related uniform polytopes and honeycombs</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=15" title="Edit section: Related uniform polytopes and honeycombs">edit</a>]</div> The regular 16-cell and <a href="/wiki/Tesseract" title="Tesseract">tesseract</a> are the regular members of a set of 15 <a href="/wiki/B4_polytope" title="B4 polytope">uniform 4-polytopes with the same B4 symmetry</a>. The 16-cell is also one of the <a href="/wiki/D4_polytope" title="D4 polytope">uniform polytopes of D4 symmetry</a>. The 16-cell is also related to the <a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">cubic honeycomb</a>, <a href="/wiki/Order-4_dodecahedral_honeycomb" title="Order-4 dodecahedral honeycomb">order-4 dodecahedral honeycomb</a>, and <a href="/wiki/Order-4_hexagonal_tiling_honeycomb" title="Order-4 hexagonal tiling honeycomb">order-4 hexagonal tiling honeycomb</a> which all have <a href="/wiki/Hexagonal_tiling_honeycomb#Polytopes_and_honeycombs_with_tetrahedral_vertex_figures" title="Hexagonal tiling honeycomb">octahedral vertex figures</a>. It belongs to the sequence of <a href="/wiki/Order-6_tetrahedral_honeycomb#Related_polytopes_and_honeycombs" title="Order-6 tetrahedral honeycomb">{3,3,p} 4-polytopes</a> which have tetrahedral cells. The sequence includes three <a href="/wiki/Regular_4-polytope" title="Regular 4-polytope">regular 4-polytopes</a> of Euclidean 4-space, the <a href="/wiki/5-cell" title="5-cell">5-cell</a> {3,3,3}, 16-cell {3,3,4}, and <a href="/wiki/600-cell" title="600-cell">600-cell</a> {3,3,5}), and the <a href="/wiki/Order-6_tetrahedral_honeycomb" title="Order-6 tetrahedral honeycomb">order-6 tetrahedral honeycomb</a> {3,3,6} of hyperbolic space. It is first in a sequence of <a href="/wiki/Tetrahedral-octahedral_honeycomb#Quasuiregular_honeycombs" title="Tetrahedral-octahedral honeycomb">quasiregular polytopes and honeycombs</a> h{4,p,q}, and a <a href="/wiki/Order-4_hexagonal_tiling_honeycomb#Quasiregular_honeycombs" title="Order-4 hexagonal tiling honeycomb">half symmetry sequence</a>, for regular forms {p,3,4}. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=16" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/24-cell" title="24-cell">24-cell</a></li> <li><a href="/wiki/4-polytope" title="4-polytope">4-polytope</a></li> <li><a href="/wiki/D4_polytope" title="D4 polytope">D4 polytope</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output 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class="navbox-list navbox-odd plainlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex" style="font-size:114%;margin:0 4em">Convex</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" 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;background:transparent;color:inherit;"><div style="padding:0px"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:100%;"><tbody><tr><td class="navbox-abovebelow" style="font-weight:bold;"><a href="/wiki/5-cell" title="5-cell">5-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Tesseract" title="Tesseract">8-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a class="mw-selflink selflink">16-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/24-cell" title="24-cell">24-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/120-cell" title="120-cell">120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/600-cell" title="600-cell">600-cell</a></td></tr><tr style="vertical-align:top"><td class="navbox-list" style="padding:0px;text-align:center;width:16%;"><div> <ul><li>{3,3,3}</li> <li>pentachoron</li> <li>4-simplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:16%;"><div> <ul><li>{4,3,3}</li> <li>tesseract</li> <li>4-cube</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:16%;"><div> <ul><li>{3,3,4}</li> <li>hexadecachoron</li> <li>4-orthoplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:16%;"><div> <ul><li>{3,4,3}</li> <li>icositetrachoron</li> <li>octaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:16%;"><div> <ul><li>{5,3,3}</li> <li>hecatonicosachoron</li> <li>dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:16%;"><div> <ul><li>{3,3,5}</li> <li>hexacosichoron</li> <li>tetraplex</li></ul> </div></td></tr></tbody></table></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd plainlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Star" style="font-size:114%;margin:0 4em">Star</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" 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;background:transparent;color:inherit;"><div style="padding:0px"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:100%;"><tbody><tr><td class="navbox-abovebelow" style="font-weight:bold;"><a href="/wiki/Icosahedral_120-cell" title="Icosahedral 120-cell">icosahedral 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Small_stellated_120-cell" title="Small stellated 120-cell">small stellated 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Great_120-cell" title="Great 120-cell">great 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Grand_120-cell" title="Grand 120-cell">grand 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Great_stellated_120-cell" title="Great stellated 120-cell">great stellated 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Grand_stellated_120-cell" title="Grand stellated 120-cell">grand stellated 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Great_grand_120-cell" title="Great grand 120-cell">great grand 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Great_icosahedral_120-cell" title="Great icosahedral 120-cell">great icosahedral 120-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Grand_600-cell" title="Grand 600-cell">grand 600-cell</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Great_grand_stellated_120-cell" title="Great grand stellated 120-cell">great grand stellated 120-cell</a></td></tr><tr style="vertical-align:top"><td class="navbox-list" style="padding:0px;text-align:center;width:10%;"><div> <ul><li>{3,5,⁠5/2⁠}</li> <li>icosaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{⁠5/2⁠,5,3}</li> <li>stellated dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{5,⁠5/2⁠,5}</li> <li>great dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{5,3,⁠5/2⁠}</li> <li>grand dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{⁠5/2⁠,3,5}</li> <li>great stellated dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{⁠5/2⁠,5,⁠5/2⁠}</li> <li>grand stellated dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{5,⁠5/2⁠,3}</li> <li>great grand dodecaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{3,⁠5/2⁠,5}</li> <li>great icosaplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{3,3,⁠5/2⁠}</li> <li>grand tetraplex</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;text-align:center;width:10%;"><div> <ul><li>{⁠5/2⁠,3,3}</li> <li>great grand stellated dodecaplex</li></ul> </div></td></tr></tbody></table></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <table class="wikitable mw-collapsible"> <tbody><tr> <th colspan="13" style="background:lightsteelblue;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-collapse navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Polytopes" title="Template:Polytopes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polytopes" title="Template talk:Polytopes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polytopes" title="Special:EditPage/Template:Polytopes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div class="navbar-ct-mini">Fundamental convex <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a> and <a href="/wiki/Uniform_polytope" title="Uniform polytope">uniform polytopes</a> in dimensions 2–10</div> </th></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Coxeter_group#Finite_Coxeter_groups" title="Coxeter group">Family</a> </th> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group">An</a> </td> <td style="background:gainsboro;"><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group">Bn</a> </td> <td style="background:gainsboro;">I2(p) / <a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group">Dn</a> </td> <td style="background:gainsboro;"><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E6</a> / <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E7</a> / <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E8</a> / <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a> / <a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a> </td> <td style="background:gainsboro;"><a href="/wiki/H4_(mathematics)" class="mw-redirect" title="H4 (mathematics)">Hn</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Regular_polygon" title="Regular polygon">Regular polygon</a> </th> <td><a href="/wiki/Equilateral_triangle" title="Equilateral triangle">Triangle</a> </td> <td><a href="/wiki/Square" title="Square">Square</a> </td> <td style="background:#f0f0e0;"><a href="/wiki/Regular_polygon" title="Regular polygon">p-gon</a> </td> <td style="background:#e0e0f0;"><a href="/wiki/Hexagon" title="Hexagon">Hexagon</a> </td> <td><a href="/wiki/Pentagon" title="Pentagon">Pentagon</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">Uniform polyhedron</a> </th> <td style="background:whitesmoke;"><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a> </td> <td style="background:whitesmoke;"><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a> • <a href="/wiki/Cube" title="Cube">Cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/Tetrahedron" title="Tetrahedron">Demicube</a> </td> <td style="background:whitesmoke;"> </td> <td style="background:whitesmoke;"><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">Dodecahedron</a> • <a href="/wiki/Regular_icosahedron" title="Regular icosahedron">Icosahedron</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_polychoron" class="mw-redirect" title="Uniform polychoron">Uniform polychoron</a> </th> <td><a href="/wiki/5-cell" title="5-cell">Pentachoron</a> </td> <td><a class="mw-selflink selflink">16-cell</a> • <a href="/wiki/Tesseract" title="Tesseract">Tesseract</a> </td> <td><a class="mw-selflink selflink">Demitesseract</a> </td> <td style="background:#e0f0e0;"><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td> <td><a href="/wiki/120-cell" title="120-cell">120-cell</a> • <a href="/wiki/600-cell" title="600-cell">600-cell</a> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_5-polytope" title="Uniform 5-polytope">Uniform 5-polytope</a> </th> <td style="background:whitesmoke;"><a href="/wiki/5-simplex" title="5-simplex">5-simplex</a> </td> <td style="background:whitesmoke;"><a href="/wiki/5-orthoplex" title="5-orthoplex">5-orthoplex</a> • <a href="/wiki/5-cube" title="5-cube">5-cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/5-demicube" title="5-demicube">5-demicube</a> </td> <td style="background:whitesmoke;"> </td> <td style="background:whitesmoke;"> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_6-polytope" title="Uniform 6-polytope">Uniform 6-polytope</a> </th> <td><a href="/wiki/6-simplex" title="6-simplex">6-simplex</a> </td> <td><a href="/wiki/6-orthoplex" title="6-orthoplex">6-orthoplex</a> • <a href="/wiki/6-cube" title="6-cube">6-cube</a> </td> <td><a href="/wiki/6-demicube" title="6-demicube">6-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_22_polytope" title="1 22 polytope">122</a> • <a href="/wiki/2_21_polytope" title="2 21 polytope">221</a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_7-polytope" title="Uniform 7-polytope">Uniform 7-polytope</a> </th> <td style="background:whitesmoke;"><a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> </td> <td style="background:whitesmoke;"><a href="/wiki/7-orthoplex" title="7-orthoplex">7-orthoplex</a> • <a href="/wiki/7-cube" title="7-cube">7-cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/7-demicube" title="7-demicube">7-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_32_polytope" title="1 32 polytope">132</a> • <a href="/wiki/2_31_polytope" title="2 31 polytope">231</a> • <a href="/wiki/3_21_polytope" title="3 21 polytope">321</a> </td> <td style="background:whitesmoke;"> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_8-polytope" title="Uniform 8-polytope">Uniform 8-polytope</a> </th> <td><a href="/wiki/8-simplex" title="8-simplex">8-simplex</a> </td> <td><a href="/wiki/8-orthoplex" title="8-orthoplex">8-orthoplex</a> • <a href="/wiki/8-cube" title="8-cube">8-cube</a> </td> <td><a href="/wiki/8-demicube" title="8-demicube">8-demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/1_42_polytope" title="1 42 polytope">142</a> • <a href="/wiki/2_41_polytope" title="2 41 polytope">241</a> • <a href="/wiki/4_21_polytope" title="4 21 polytope">421</a> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_9-polytope" title="Uniform 9-polytope">Uniform 9-polytope</a> </th> <td style="background:whitesmoke;"><a href="/wiki/9-simplex" title="9-simplex">9-simplex</a> </td> <td style="background:whitesmoke;"><a href="/wiki/9-orthoplex" title="9-orthoplex">9-orthoplex</a> • <a href="/wiki/9-cube" title="9-cube">9-cube</a> </td> <td style="background:whitesmoke;"><a href="/wiki/9-demicube" title="9-demicube">9-demicube</a> </td> <td style="background:whitesmoke;"> </td> <td style="background:whitesmoke;"> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;"><a href="/wiki/Uniform_10-polytope" title="Uniform 10-polytope">Uniform 10-polytope</a> </th> <td><a href="/wiki/10-simplex" title="10-simplex">10-simplex</a> </td> <td><a href="/wiki/10-orthoplex" title="10-orthoplex">10-orthoplex</a> • <a href="/wiki/10-cube" title="10-cube">10-cube</a> </td> <td><a href="/wiki/10-demicube" title="10-demicube">10-demicube</a> </td> <td> </td> <td> </td></tr> <tr style="text-align:center;"> <th style="background:gainsboro;">Uniform n-<a href="/wiki/Polytope" title="Polytope">polytope</a> </th> <td style="background:whitesmoke;">n-<a href="/wiki/Simplex" title="Simplex">simplex</a> </td> <td style="background:whitesmoke;">n-<a href="/wiki/Cross-polytope" title="Cross-polytope">orthoplex</a> • n-<a href="/wiki/Hypercube" title="Hypercube">cube</a> </td> <td style="background:whitesmoke;">n-<a href="/wiki/Demihypercube" title="Demihypercube">demicube</a> </td> <td style="background:#f0e0e0;"><a href="/wiki/Uniform_1_k2_polytope" title="Uniform 1 k2 polytope">1k2</a> • <a href="/wiki/Uniform_2_k1_polytope" title="Uniform 2 k1 polytope">2k1</a> • <a href="/wiki/Uniform_k_21_polytope" title="Uniform k 21 polytope">k21</a> </td> <td style="background:whitesmoke;">n-<a href="/wiki/Pentagonal_polytope" title="Pentagonal polytope">pentagonal polytope</a> </td></tr> <tr style="text-align:center;"> <th colspan="13" style="background:gainsboro;">Topics: <a href="/wiki/Polytope_families" title="Polytope families">Polytope families</a> • <a href="/wiki/Regular_polytope" title="Regular polytope">Regular polytope</a> • <a href="/wiki/List_of_regular_polytopes_and_compounds" class="mw-redirect" title="List of regular polytopes and compounds">List of regular polytopes and compounds</a> </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=17" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-lower-alpha" style="column-width: 40em;"> <ol class="references"> <li id="cite_note-polytopes_ordered_by_size_and_complexity-6"><a href="#cite_ref-polytopes_ordered_by_size_and_complexity_6-0">^</a> The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content<a href="#cite_note-FOOTNOTECoxeter1973292–293Table_I(ii):_The_sixteen_regular_polytopes_{''p,q,r''}_in_four_dimensions-5">[5]</a> within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing <a href="#As_a_configuration">configuration matrices</a> or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 16-cell is the 8-point 4-polytope: second in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> There are 2 and only 2 16-cells inscribed in the 8-cell (tesseract), 3 and only 3 16-cells inscribed in the 24-cell, 75 distinct 16-cells (but only 15 disjoint 16-cells) inscribed in the 600-cell, and 675 distinct 16-cells (but only 75 disjoint 16-cells) inscribed in the 120-cell. </li> <li id="cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">^ <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-0">a</a> <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-1">b</a> <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-2">c</a> <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-3">d</a> <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-4">e</a> <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-5">f</a> <a href="#cite_ref-six_orthogonal_planes_of_the_Cartesian_basis_8-6">g</a> In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin. </li> <li id="cite_note-octahedral_pyramid-9">^ <a href="#cite_ref-octahedral_pyramid_9-0">a</a> <a href="#cite_ref-octahedral_pyramid_9-1">b</a> <a href="#cite_ref-octahedral_pyramid_9-2">c</a> Each vertex in the 16-cell is the apex of an <a href="/wiki/Octahedral_pyramid" title="Octahedral pyramid">octahedral pyramid</a>, the base of which is the octahedron formed by the 6 other vertices to which the apex is connected by edges. The 16-cell can be deconstructed (four different ways) into two octahedral pyramids by cutting it in half through one of its four octahedral central hyperplanes. Looked at from inside the curved 3 dimensional volume of its boundary surface of 16 face-bonded tetrahedra, the 16-cell's vertex figure is an octahedron. In 4 dimensions, the vertex octahedron is actually an octahedral pyramid. The apex of the octahedral pyramid (the vertex where the 6 edges meet) is not actually at the center of the octahedron: it is displaced radially outwards in the fourth dimension, out of the hyperplane defined by the octahedron's 6 vertices. The 6 edges around the vertex make an orthogonal 3-axis cross in 3 dimensions (and in the <a href="/wiki/Octahedral_pyramid" title="Octahedral pyramid">3-dimensional projection of the 4-pyramid</a>), but the 3 lines are actually bent 90 degrees in the fourth dimension where they meet in an apex. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> The boundary surface of a 16-cell is a finite 3-dimensional space consisting of 16 tetrahedra arranged face-to-face (four around one). It is a closed, tightly curved (non-Euclidean) 3-space, within which we can move straight through 4 tetrahedra in any direction and arrive back in the tetrahedron where we started. We can visualize moving around inside this tetrahedral <a href="/wiki/Jungle_gym" title="Jungle gym">jungle gym</a>, climbing from one tetrahedron into another on its 24 struts (its edges), and never being able to get out (or see out) of the 16 tetrahedra no matter what direction we go (or look). We are always on (or in) the surface of the 16-cell, never inside the 16-cell itself (nor outside it). We can see that the 6 edges around each vertex radiate symmetrically in 3 dimensions and form an orthogonal 3-axis cross, just as the radii of an octahedron do (so we say the vertex figure of the 16-cell is the octahedron).<a href="#cite_note-octahedral_pyramid-9">[d]</a> </li> <li id="cite_note-completely_orthogonal_planes-11">^ <a href="#cite_ref-completely_orthogonal_planes_11-0">a</a> <a href="#cite_ref-completely_orthogonal_planes_11-1">b</a> <a href="#cite_ref-completely_orthogonal_planes_11-2">c</a> Two flat planes A and B of a Euclidean space of four dimensions are called completely orthogonal if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O. A and B are perpendicular and <a href="#Octahedral_dipyramid">Clifford parallel</a>.<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> </li> <li id="cite_note-Clifford_parallel_great_squares-13">^ <a href="#cite_ref-Clifford_parallel_great_squares_13-0">a</a> <a href="#cite_ref-Clifford_parallel_great_squares_13-1">b</a> <a href="#cite_ref-Clifford_parallel_great_squares_13-2">c</a> <a href="#cite_ref-Clifford_parallel_great_squares_13-3">d</a> Completely orthogonal great squares are non-intersecting and rotate independently because the great circles on which their vertices lie are <a href="/wiki/Clifford_parallel" title="Clifford parallel">Clifford parallel</a>.<a href="#cite_note-Clifford_parallels-20">[l]</a> They are √2 apart at each pair of nearest vertices (and in the 16-cell all the pairs except antipodal pairs are nearest). The two squares cannot intersect at all because they lie in planes which intersect at only one point: the center of the 16-cell.<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> Because they are perpendicular and share a common center, the two squares are obviously not parallel and separate in the usual way of parallel squares in 3 dimensions; rather they are connected like adjacent square links in a chain, each passing through the other without intersecting at any points, forming a <a href="/wiki/Hopf_link" title="Hopf link">Hopf link</a>. </li> <li id="cite_note-octahedral_hyperplanes-14">^ <a href="#cite_ref-octahedral_hyperplanes_14-0">a</a> <a href="#cite_ref-octahedral_hyperplanes_14-1">b</a> Three great squares meet at each vertex (and at its opposite vertex) in the 16-cell. Each of them has a different completely orthogonal square.<a href="#cite_note-completely_orthogonal_planes-11">[f]</a> Thus there are three great squares completely orthogonal to each vertex and its opposite vertex (each axis). They form an octahedron (a central hyperplane). Every axis line in the 16-cell is completely orthogonal to a central octahedron hyperplane, as every great square plane is completely orthogonal to another great square plane.<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> The axis and the octahedron intersect only at one point (the center of the 16-cell), as each pair of completely orthogonal great squares intersects only at one point (the center of the 16-cell). Each central octahedron is also the octahedral vertex figure of two of the eight vertices: the two on its completely orthogonal axis. </li> <li id="cite_note-completely_orthogonal_great_squares-15"><a href="#cite_ref-completely_orthogonal_great_squares_15-0">^</a> The three incompletely orthogonal great squares which intersect at each vertex of the 16-cell form the vertex's octahedral <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a>.<a href="#cite_note-octahedral_pyramid-9">[d]</a> Any two of them, together with the completely orthogonal square of the third, also form an octahedron: a central octahedral hyperplane.<a href="#cite_note-octahedral_hyperplanes-14">[h]</a> In the 16-cell, each octahedral vertex figure is also a central octahedral hyperplane. </li> <li id="cite_note-vertex_and_central_octahedra-16"><a href="#cite_ref-vertex_and_central_octahedra_16-0">^</a> Each great square vertex is √2 distant from two of the square's other vertices, and √4 distant from its opposite vertex. The other four vertices of the 16-cell (also √2 distant) are the vertices of the square's completely orthogonal square.<a href="#cite_note-Clifford_parallel_great_squares-13">[g]</a> Each 16-cell vertex is a vertex of three orthogonal great squares which intersect there. Each of them has a different completely orthogonal square. Thus there are three great squares completely orthogonal to each vertex: squares that the vertex is not part of.<a href="#cite_note-completely_orthogonal_great_squares-15">[i]</a> </li> <li id="cite_note-completely_orthogonal_Clifford_parallels_are_special-19">^ <a href="#cite_ref-completely_orthogonal_Clifford_parallels_are_special_19-0">a</a> <a href="#cite_ref-completely_orthogonal_Clifford_parallels_are_special_19-1">b</a> <a href="#cite_ref-completely_orthogonal_Clifford_parallels_are_special_19-2">c</a> <a href="#cite_ref-completely_orthogonal_Clifford_parallels_are_special_19-3">d</a> Each great square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal<a href="#cite_note-completely_orthogonal_planes-11">[f]</a> to only one of them. Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal. There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a>. A pair of isoclinic (Clifford parallel) planes is either a left pair or a right pair unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).<a href="#cite_note-FOOTNOTEKimRote20167–8§&nbsp;6_Angles_between_two_Planes_in_4-Space-49">[20]</a> Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. Because planes separated by a 90° isoclinic rotation are 180° apart, the plane to the left and the plane to the right are the same plane.<a href="#cite_note-exchange_of_completely_orthogonal_planes-27">[r]</a> </li> <li id="cite_note-Clifford_parallels-20">^ <a href="#cite_ref-Clifford_parallels_20-0">a</a> <a href="#cite_ref-Clifford_parallels_20-1">b</a> <a href="#cite_ref-Clifford_parallels_20-2">c</a> <a href="/wiki/Clifford_parallel" title="Clifford parallel">Clifford parallels</a> are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.<a href="#cite_note-FOOTNOTETyrrellSemple19715–6§&nbsp;3._Clifford's_original_definition_of_parallelism-17">[7]</a> A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a>.<a href="#cite_note-FOOTNOTEKimRote20167–10§&nbsp;6._Angles_between_two_Planes_in_4-Space-18">[8]</a> In the 16-cell the corresponding vertices of completely orthogonal great circle squares are all √2 apart, so these squares are Clifford parallel polygons.<a href="#cite_note-completely_orthogonal_Clifford_parallels_are_special-19">[k]</a> Note that only the vertices of the great squares (the points on the great circle) are √2 apart; points on the edges of the squares (on chords of the circle) are closer together. </li> <li id="cite_note-long_diagonal_of_the_4-cube-21"><a href="#cite_ref-long_diagonal_of_the_4-cube_21-0">^</a> Opposite vertices in a unit-radius 4-polytope correspond to the opposite vertices of an 8-cell hypercube (tesseract). The long diagonal of this <a href="/wiki/Tesseract#Radial_equilateral_symmetry" title="Tesseract">radially equilateral 4-cube</a> is √4. In a 90° isoclinic rotation each vertex of the 16-cell is displaced to its antipodal vertex, traveling along a helical geodesic arc of length 𝝅 (180°), to a vertex √4 away along the long diameter of the unit-radius 4-polytope (16-cell or tesseract), the same total displacement as if it had been displaced √1 four times by traveling along a path of four successive orthogonal edges of the tesseract. </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> There are six different two-edge paths connecting a pair of antipodal vertices along the edges of a great square. The left isoclinic rotation runs diagonally between three of them, and the right isoclinic rotation runs diagonally between the other three. These diagonals are the straight lines (geodesics) connecting opposite vertices of face-bonded tetrahedral cells in the left-handed <a href="#Helical_construction">eight-cell ring</a> and the right-handed eight-cell ring, respectively. </li> <li id="cite_note-isocline-23">^ <a href="#cite_ref-isocline_23-0">a</a> <a href="#cite_ref-isocline_23-1">b</a> <a href="#cite_ref-isocline_23-2">c</a> In the 16-cell, two antipodal vertices are opposite vertices of two face-bonded tetrahedral cells. The two antipodal vertices are connected by (three different) two-edge great circle paths along edges of the tetrahedral cells, by various three-edge paths, and by four-edge paths on isoclines and Petrie polygons. <a href="#cite_note-Möbius_circle-24">[p]</a> </li> <li id="cite_note-Möbius_circle-24">^ <a href="#cite_ref-Möbius_circle_24-0">a</a> <a href="#cite_ref-Möbius_circle_24-1">b</a> <a href="#cite_ref-Möbius_circle_24-2">c</a> <a href="#cite_ref-Möbius_circle_24-3">d</a> An isocline is a circle of special kind corresponding to a pair of <a href="/wiki/Villarceau_circle" class="mw-redirect" title="Villarceau circle">Villarceau circles</a> linked in a <a href="/wiki/M%C3%B6bius_loop" class="mw-redirect" title="Möbius loop">Möbius loop</a>. It curves through four dimensions instead of just two. All ordinary circles have a 2𝝅 circumference, but the 16-cell's isocline is a circle with an 4𝝅 circumference (over eight 90° chords). An isocline is a circle that does not lie in a plane, but to avoid confusion we always refer to it as an isocline and reserve the term circle for an ordinary circle in the plane. </li> <li id="cite_note-isoclinic_rotation-25">^ <a href="#cite_ref-isoclinic_rotation_25-0">a</a> <a href="#cite_ref-isoclinic_rotation_25-1">b</a> In an isoclinic rotation, all 6 orthogonal planes are displaced in two orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted sideways by that same angle. An isoclinic displacement (also known as a <a href="/wiki/William_Kingdon_Clifford" title="William Kingdon Clifford">Clifford</a> displacement) is 4-dimensionally diagonal. Points are displaced an equal distance in four orthogonal directions at once, and displaced a total <a href="/wiki/Pythagorean_distance#Higher_dimensions" class="mw-redirect" title="Pythagorean distance">Pythagorean distance</a> equal to the square root of four times the square of that distance. All vertices of a regular 4-polytope are displaced to a vertex at least two edge lengths away. For example, when the unit-radius 16-cell rotates isoclinically 90° in a great square invariant plane, it also rotates 90° in the completely orthogonal great square invariant plane.<a href="#cite_note-six_orthogonal_planes_of_the_Cartesian_basis-8">[c]</a> The great square plane also tilts sideways 90° to occupy its completely orthogonal plane. (By isoclinic symmetry, every great square rotates 90° and tilts sideways 90° into its completely orthogonal plane.) Each vertex (in every great square) is displaced to its antipodal vertex, at a distance of √1 in each of four orthogonal directions, a total distance of √4.<a href="#cite_note-long_diagonal_of_the_4-cube-21">[m]</a> The original and displaced vertex are two edge lengths apart by three<a href="#cite_note-22">[n]</a> different paths along two edges of a great square. But the <a href="#Helical_construction">isocline</a> (the helical arc the vertex follows during the isoclinic rotation) does not run along edges: it runs between these different edge-paths diagonally, on a geodesic (shortest arc) between the original and displaced vertices.<a href="#cite_note-isocline-23">[o]</a> This isoclinic geodesic arc is not a segment of an ordinary great circle; it does not lie in the plane of any great square. It is a helical 180° arc that bends in a circle in two completely orthogonal planes at once. This <a href="/wiki/M%C3%B6bius_loop" class="mw-redirect" title="Möbius loop">Möbius circle</a> does not lie in any plane or intersect any vertices between the original and the displaced vertex.<a href="#cite_note-Möbius_circle-24">[p]</a> </li> <li id="cite_note-exchange_of_completely_orthogonal_planes-27">^ <a href="#cite_ref-exchange_of_completely_orthogonal_planes_27-0">a</a> <a href="#cite_ref-exchange_of_completely_orthogonal_planes_27-1">b</a> <a href="#cite_ref-exchange_of_completely_orthogonal_planes_27-2">c</a> The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 16-cell, all 6 orthogonal planes rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel)<a href="#cite_note-Clifford_parallels-20">[l]</a> plane.<a href="#cite_note-FOOTNOTEKimRote20168–10Relations_to_Clifford_Parallelism-26">[9]</a> The corresponding vertices of the two completely orthogonal great squares are √4 (180°) apart; the great squares (Clifford parallel polytopes) are √4 (180°) apart; but the two completely orthogonal planes are 90° apart, in the two orthogonal angles that separate them. If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great square returns to its original plane, but in a different orientation (axes swapped): it has been turned "upside down" on the surface of the 16-cell (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation. </li> <li id="cite_note-characteristic_orthoscheme-30"><a href="#cite_ref-characteristic_orthoscheme_30-0">^</a> An <a href="/wiki/Orthoscheme" class="mw-redirect" title="Orthoscheme">orthoscheme</a> is a <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> irregular <a href="/wiki/Simplex" title="Simplex">simplex</a> with <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own <a href="/wiki/Facet_(geometry)" title="Facet (geometry)">facets</a> (its mirror walls). Every regular polytope can be dissected radially into instances of its <a href="/wiki/Orthoscheme#Characteristic_simplex_of_the_general_regular_polytope" class="mw-redirect" title="Orthoscheme">characteristic orthoscheme</a> surrounding its center. The characteristic orthoscheme has the shape described by the same <a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter-Dynkin diagram</a> as the regular polytope without the generating point ring. </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> A regular polytope of dimension k has a characteristic k-orthoscheme, and also a characteristic (k-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.<a href="#cite_note-FOOTNOTECoxeter1973130§&nbsp;7.6-31">[12]</a> The interior tetrahedra and triangles thus formed will also be orthoschemes. </li> <li id="cite_note-reversed_greek_symbols-36">^ <a href="#cite_ref-reversed_greek_symbols_36-0">a</a> <a href="#cite_ref-reversed_greek_symbols_36-1">b</a> (<a href="#CITEREFCoxeter1973">Coxeter 1973</a>) uses the greek letter 𝝓 (phi) to represent one of the three characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle. </li> <li id="cite_note-characteristic_radii-37"><a href="#cite_ref-characteristic_radii_37-0">^</a> The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets. </li> <li id="cite_note-only_one_disjoint_pair_of_eight-cell_rings-40">^ <a href="#cite_ref-only_one_disjoint_pair_of_eight-cell_rings_40-0">a</a> <a href="#cite_ref-only_one_disjoint_pair_of_eight-cell_rings_40-1">b</a> The 16-cell can be constructed from two cell-disjoint eight-cell rings in three different ways; it has three orientations of its pair of rings. Each orientation "contains" a distinct left-right pair of isoclinic rotations, and also a pair of completely orthogonal great squares (Clifford parallel fibers), so each orientation is a discrete <a href="/wiki/Hopf_fibration" title="Hopf fibration">fibration</a> of the 16-cell. Each eight-cell ring contains three axial octagrams which have different orientations (they exchange roles) in the three discrete fibrations and six distinct isoclinic rotations (three left and three right) through the cell rings. Three octagrams (of different colors) can be seen in the illustration of a single cell ring, one in the role of Petrie polygon, one as the right isocline, and one as the left isocline. Because each octagram plays three roles, there are exactly six distinct isoclines in the 16-cell, not 18. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> All five views are the same orthogonal projection of the 16-cell into the same plane (a circular cross-section of the eight-cell ring cylinder), looking along the central axis of the cut ring cylinder pictured above, from one end of the cylinder. The only difference is which √2 edges and √4 chords are omitted for focus. The different colors of √2 edges appear to be of different lengths because they are oblique to the viewer at different angles. Vertices are numbered 1 (top) through 8 in counterclockwise order. </li> <li id="cite_note-isocline_curve-44">^ <a href="#cite_ref-isocline_curve_44-0">a</a> <a href="#cite_ref-isocline_curve_44-1">b</a> Each isocline has the eight √2 chords of its edge-path, and also four √4 diameter chords that connect every fourth vertex on the hexagram{8/3}. Antipodal vertices also have a twisted path of four mutually orthogonal √2 edges connecting them. Between antipodal vertices, the isocline curves smoothly around in a helix over the √2 chords of its edge-path, hitting three intervening vertices. Each √2 edge is an edge of a great square, that is completely orthogonal to another great square, in which the √4 chord is a diagonal. </li> <li id="cite_note-two_special_cases-45">^ <a href="#cite_ref-two_special_cases_45-0">a</a> <a href="#cite_ref-two_special_cases_45-1">b</a> For another example of the left and right isoclines of a rotation visiting the same set of vertices, see the <a href="/wiki/5-cell#Geodesics_and_rotations" title="5-cell">characteristic isoclinic rotation of the 5-cell</a>. Although in these two special cases left and right isoclines of the same rotation visit the same set of vertices, they still take very different rotational paths because they visit the same vertices in different sequences. </li> <li id="cite_note-counter-rotating_double_helix-46">^ <a href="#cite_ref-counter-rotating_double_helix_46-0">a</a> <a href="#cite_ref-counter-rotating_double_helix_46-1">b</a> Except in the 5-cell and 16-cell,<a href="#cite_note-two_special_cases-45">[z]</a> a pair of left and right isocline circles have disjoint vertices: the left and right isocline helices are non-intersecting parallels but counter-rotating, forming a special kind of double helix which cannot occur in three dimensions (where counter-rotating helices of the same radius must intersect). </li> <li id="cite_note-each_16-cell_isocline_reaches_all_8_vertices-47">^ <a href="#cite_ref-each_16-cell_isocline_reaches_all_8_vertices_47-0">a</a> <a href="#cite_ref-each_16-cell_isocline_reaches_all_8_vertices_47-1">b</a> In the 16-cell each single isocline winds through all 8 vertices: an entire <a href="/wiki/Hopf_fibration" title="Hopf fibration">fibration</a> of two completely orthogonal great squares.<a href="#cite_note-completely_orthogonal_Clifford_parallels_are_special-19">[k]</a> The 5-cell and the 16-cell are the only regular 4-polytopes where each discrete fibration has just one isocline fiber.<a href="#cite_note-counter-rotating_double_helix-46">[aa]</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> The left and right isoclines intersect each other at every vertex. They are different sequences of the same set of 8 vertices. With respect only to the set of 4 vertex pairs which are √2 apart, they can be considered to be Clifford parallel. With respect only to the set of 4 vertex pairs which are √4 apart, they can be considered to be completely orthogonal.<a href="#cite_note-completely_orthogonal_Clifford_parallels_are_special-19">[k]</a> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> This is atypical for isoclinic rotations generally; normally both the left and right isoclines do not occur at the same vertex: there are two disjoint sets of vertices reachable only by the left or right rotation respectively.<a href="#cite_note-counter-rotating_double_helix-46">[aa]</a> The left and right isoclines of the 16-cell form a very special double helix: unusual not just because it is circular, but because its different left and right helices twist around each other through the same set of antipodal vertices,<a href="#cite_note-each_16-cell_isocline_reaches_all_8_vertices-47">[ab]</a> not through the two disjoint subsets of antipodal vertices, as the isocline pairs do in most isoclinic rotations found in nature.<a href="#cite_note-two_special_cases-45">[z]</a> Isoclinic rotations in completely orthogonal invariant planes are special.<a href="#cite_note-completely_orthogonal_Clifford_parallels_are_special-19">[k]</a> To see how and why they are special, visualize two completely orthogonal invariant planes of rotation, each rotating by some rotation angle and tilting sideways by the same rotation angle into a different plane entirely.<a href="#cite_note-isoclinic_rotation-25">[q]</a> Only when the rotation angle is 90°, that different plane in which the tilting invariant plane lands will be the completely orthogonal invariant plane itself. The destination plane of the rotation is the completely orthogonal invariant plane. The 90° isoclinic rotation is the only rotation which takes the completely orthogonal invariant planes to each other.<a href="#cite_note-exchange_of_completely_orthogonal_planes-27">[r]</a> This reciprocity is the reason both left and right rotations go to the same place. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=18" title="Edit section: Citations">edit</a>]</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-FOOTNOTECoxeter1973141§&nbsp;7-x._Historical_remarks-1"><a href="#cite_ref-FOOTNOTECoxeter1973141§&nbsp;7-x._Historical_remarks_1-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 141, § 7-x. Historical remarks. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">N.W. Johnson</a>: Geometries and Transformations, (2018) <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-10340-5" title="Special:BookSources/978-1-107-10340-5">978-1-107-10340-5</a> Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249 </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Matila Ghyka, The Geometry of Art and Life (1977), p.68 </li> <li id="cite_note-FOOTNOTECoxeter1973120=121§&nbsp;7.2._See_illustration_Fig_7.2B-4"><a href="#cite_ref-FOOTNOTECoxeter1973120=121§&nbsp;7.2._See_illustration_Fig_7.2B_4-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 120=121, § 7.2. See illustration Fig 7.2B. </li> <li id="cite_note-FOOTNOTECoxeter1973292–293Table_I(ii):_The_sixteen_regular_polytopes_{''p,q,r''}_in_four_dimensions-5"><a href="#cite_ref-FOOTNOTECoxeter1973292–293Table_I(ii):_The_sixteen_regular_polytopes_{''p,q,r''}_in_four_dimensions_5-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 292–293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius. </li> <li id="cite_note-FOOTNOTEKimRote20166§&nbsp;5._Four-Dimensional_Rotations-12"><a href="#cite_ref-FOOTNOTEKimRote20166§&nbsp;5._Four-Dimensional_Rotations_12-0">^</a> <a href="#CITEREFKimRote2016">Kim & Rote 2016</a>, p. 6, § 5. Four-Dimensional Rotations. </li> <li id="cite_note-FOOTNOTETyrrellSemple19715–6§&nbsp;3._Clifford's_original_definition_of_parallelism-17"><a href="#cite_ref-FOOTNOTETyrrellSemple19715–6§&nbsp;3._Clifford's_original_definition_of_parallelism_17-0">^</a> <a href="#CITEREFTyrrellSemple1971">Tyrrell & Semple 1971</a>, pp. 5–6, § 3. Clifford's original definition of parallelism. </li> <li id="cite_note-FOOTNOTEKimRote20167–10§&nbsp;6._Angles_between_two_Planes_in_4-Space-18"><a href="#cite_ref-FOOTNOTEKimRote20167–10§&nbsp;6._Angles_between_two_Planes_in_4-Space_18-0">^</a> <a href="#CITEREFKimRote2016">Kim & Rote 2016</a>, pp. 7–10, § 6. Angles between two Planes in 4-Space. </li> <li id="cite_note-FOOTNOTEKimRote20168–10Relations_to_Clifford_Parallelism-26"><a href="#cite_ref-FOOTNOTEKimRote20168–10Relations_to_Clifford_Parallelism_26-0">^</a> <a href="#CITEREFKimRote2016">Kim & Rote 2016</a>, pp. 8–10, Relations to Clifford Parallelism. </li> <li id="cite_note-FOOTNOTECoxeter1973121§&nbsp;7.21._See_illustration_Fig_7.2B-28"><a href="#cite_ref-FOOTNOTECoxeter1973121§&nbsp;7.21._See_illustration_Fig_7.2B_28-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 121, § 7.21. See illustration Fig 7.2B: "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16fb7d2f7d9b9f310820b4e110b084003aa80fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{4}}"> is a four-dimensional dipyramid based on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1001bcd6968314a965726f5dd193d1b11ada59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{3}}"> (with its two apices in opposite directions along the fourth dimension)." </li> <li id="cite_note-FOOTNOTETyrrellSemple1971-29"><a href="#cite_ref-FOOTNOTETyrrellSemple1971_29-0">^</a> <a href="#CITEREFTyrrellSemple1971">Tyrrell & Semple 1971</a>. </li> <li id="cite_note-FOOTNOTECoxeter1973130§&nbsp;7.6-31"><a href="#cite_ref-FOOTNOTECoxeter1973130§&nbsp;7.6_31-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 130, § 7.6; "simplicial subdivision". </li> <li id="cite_note-FOOTNOTECoxeter1973292–293Table_I(ii);_"16-cell,_𝛽4"-33"><a href="#cite_ref-FOOTNOTECoxeter1973292–293Table_I(ii);_"16-cell,_𝛽4"_33-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 292–293, Table I(ii); "16-cell, 𝛽4". </li> <li id="cite_note-FOOTNOTECoxeter1973139§&nbsp;7.9_The_characteristic_simplex-34"><a href="#cite_ref-FOOTNOTECoxeter1973139§&nbsp;7.9_The_characteristic_simplex_34-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 139, § 7.9 The characteristic simplex. </li> <li id="cite_note-FOOTNOTECoxeter1973290Table_I(ii);_"dihedral_angles"-35"><a href="#cite_ref-FOOTNOTECoxeter1973290Table_I(ii);_"dihedral_angles"_35-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 290, Table I(ii); "dihedral angles". </li> <li id="cite_note-FOOTNOTECoxeter197045Table_2:_Reflexible_honeycombs_and_their_groups-38"><a href="#cite_ref-FOOTNOTECoxeter197045Table_2:_Reflexible_honeycombs_and_their_groups_38-0">^</a> <a href="#CITEREFCoxeter1970">Coxeter 1970</a>, p. 45, Table 2: Reflexible honeycombs and their groups; Honeycomb [3,3,4]4 is a tiling of the 3-sphere by 2 rings of 8 tetrahedral cells. </li> <li id="cite_note-FOOTNOTEBanchoff2013-39"><a href="#cite_ref-FOOTNOTEBanchoff2013_39-0">^</a> <a href="#CITEREFBanchoff2013">Banchoff 2013</a>. </li> <li id="cite_note-FOOTNOTECoxeter1973292–293Table_I(ii);_24-cell_''h1''-42"><a href="#cite_ref-FOOTNOTECoxeter1973292–293Table_I(ii);_24-cell_''h1''_42-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 292–293, Table I(ii); 24-cell h1. </li> <li id="cite_note-FOOTNOTECoxeter1973292–293Table_I(ii);_24-cell_''h2''-43"><a href="#cite_ref-FOOTNOTECoxeter1973292–293Table_I(ii);_24-cell_''h2''_43-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, pp. 292–293, Table I(ii); 24-cell h2. </li> <li id="cite_note-FOOTNOTEKimRote20167–8§&nbsp;6_Angles_between_two_Planes_in_4-Space-49"><a href="#cite_ref-FOOTNOTEKimRote20167–8§&nbsp;6_Angles_between_two_Planes_in_4-Space_49-0">^</a> <a href="#CITEREFKimRote2016">Kim & Rote 2016</a>, pp. 7–8, § 6 Angles between two Planes in 4-Space; Left and Right Pairs of Isoclinic Planes. </li> <li id="cite_note-FOOTNOTECoxeter1973293-51"><a href="#cite_ref-FOOTNOTECoxeter1973293_51-0">^</a> <a href="#CITEREFCoxeter1973">Coxeter 1973</a>, p. 293. </li> <li id="cite_note-FOOTNOTECoxeter199130,_47-52"><a href="#cite_ref-FOOTNOTECoxeter199130,_47_52-0">^</a> <a href="#CITEREFCoxeter1991">Coxeter 1991</a>, pp. 30, 47. </li> <li id="cite_note-FOOTNOTECoxeterShephard1992-53"><a href="#cite_ref-FOOTNOTECoxeterShephard1992_53-0">^</a> <a href="#CITEREFCoxeterShephard1992">Coxeter & Shephard 1992</a>. </li> <li id="cite_note-FOOTNOTECoxeter1991108-54"><a href="#cite_ref-FOOTNOTECoxeter1991108_54-0">^</a> <a href="#CITEREFCoxeter1991">Coxeter 1991</a>, p. 108. </li> <li id="cite_note-FOOTNOTECoxeter1991114-55"><a href="#cite_ref-FOOTNOTECoxeter1991114_55-0">^</a> <a href="#CITEREFCoxeter1991">Coxeter 1991</a>, p. 114. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=19" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Thorold_Gosset" title="Thorold Gosset">T. Gosset</a>: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900</li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">H.S.M. Coxeter</a>: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1973" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1973). <a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)">Regular Polytopes</a> (3rd ed.). New York: Dover.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Polytopes&rft.place=New+York&rft.edition=3rd&rft.pub=Dover&rft.date=1973&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1991" class="citation book cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Regular+Complex+Polytopes&rft.place=Cambridge&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1991&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li> <li>Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-01003-6" title="Special:BookSources/978-0-471-01003-6">978-0-471-01003-6</a> <a rel="nofollow" class="external text" href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley</a> <ul><li>(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]</li> <li>(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]</li> <li>(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeterShephard1992" class="citation journal cs1"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a>; Shephard, G.C. (1992). "Portraits of a family of complex polytopes". Leonardo. 25 (3/4): 239–244. <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%2F1575843">10.2307/1575843</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/1575843">1575843</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:124245340">124245340</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Leonardo&rft.atitle=Portraits+of+a+family+of+complex+polytopes&rft.volume=25&rft.issue=3%2F4&rft.pages=239-244&rft.date=1992&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124245340%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1575843%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1575843&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rft.au=Shephard%2C+G.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1970" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H.S.M.</a> (1970), "Twisted Honeycombs", Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 4, Providence, Rhode Island: American Mathematical Society</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Conference+Board+of+the+Mathematical+Sciences+Regional+Conference+Series+in+Mathematics&rft.atitle=Twisted+Honeycombs&rft.volume=4&rft.date=1970&rft.aulast=Coxeter&rft.aufirst=H.S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li></ul></li> <li><a href="/wiki/John_Horton_Conway" title="John Horton Conway">John H. Conway</a>, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-220-5" title="Special:BookSources/978-1-56881-220-5">978-1-56881-220-5</a> (Chapter 26. pp. 409: Hemicubes: 1n1)</li> <li><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">Norman Johnson</a> Uniform Polytopes, Manuscript (1991) <ul><li>N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimRote2016" class="citation arxiv cs1">Kim, Heuna; Rote, Günter (2016). "Congruence Testing of Point Sets in 4 Dimensions". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1603.07269">1603.07269</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CG">cs.CG</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Congruence+Testing+of+Point+Sets+in+4+Dimensions&rft.date=2016&rft_id=info%3Aarxiv%2F1603.07269&rft.aulast=Kim&rft.aufirst=Heuna&rft.au=Rote%2C+G%C3%BCnter&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTyrrellSemple1971" class="citation book cs1">Tyrrell, J. A.; Semple, J.G. (1971). <a rel="nofollow" class="external text" href="https://archive.org/details/generalizedcliff0000tyrr">Generalized Clifford parallelism</a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-08042-8" title="Special:BookSources/0-521-08042-8"><bdi>0-521-08042-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Clifford+parallelism&rft.pub=Cambridge+University+Press&rft.date=1971&rft.isbn=0-521-08042-8&rft.aulast=Tyrrell&rft.aufirst=J.+A.&rft.au=Semple%2C+J.G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeneralizedcliff0000tyrr&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBanchoff2013" class="citation book cs1">Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). <a rel="nofollow" class="external text" href="https://archive.org/details/shapingspaceexpl00sene">Shaping Space</a>. Springer New York. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/shapingspaceexpl00sene/page/n249">257</a>–266. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-92714-5_20">10.1007/978-0-387-92714-5_20</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-92713-8" title="Special:BookSources/978-0-387-92713-8"><bdi>978-0-387-92713-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Torus+Decompostions+of+Regular+Polytopes+in+4-space&rft.btitle=Shaping+Space&rft.pages=257-266&rft.pub=Springer+New+York&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-0-387-92714-5_20&rft.isbn=978-0-387-92713-8&rft.aulast=Banchoff&rft.aufirst=Thomas+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshapingspaceexpl00sene&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=16-cell&action=edit&section=20" title="Edit section: External links">edit</a>]</div> <ul><li><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/16-Cell.html">"16-Cell"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</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=16-Cell&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2F16-Cell.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.polytope.de/c16.html">Der 16-Zeller (16-cell)</a> Marco Möller's Regular polytopes in R4 (German)</li> <li><a rel="nofollow" class="external text" href="http://eusebeia.dyndns.org/4d/16-cell.html">Description and diagrams of 16-cell projections</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlitzing" class="citation web cs1">Klitzing, Richard. <a rel="nofollow" class="external text" href="https://bendwavy.org/klitzing/dimensions/polychora.htm">"4D uniform polytopes (polychora) x3o3o4o – hex"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=4D+uniform+polytopes+%28polychora%29+x3o3o4o+%E2%80%93+hex&rft.aulast=Klitzing&rft.aufirst=Richard&rft_id=https%3A%2F%2Fbendwavy.org%2Fklitzing%2Fdimensions%2Fpolychora.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3A16-cell" class="Z3988"></li></ul>    </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=16-cell&oldid=1257023309">https://en.wikipedia.org/w/index.php?title=16-cell&oldid=1257023309</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Regular_4-polytopes" title="Category:Regular 4-polytopes">Regular 4-polytopes</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_cleanup_from_May_2024" title="Category:Articles needing cleanup from May 2024">Articles needing cleanup from May 2024</a></li><li><a href="/wiki/Category:All_pages_needing_cleanup" title="Category:All pages needing cleanup">All pages needing cleanup</a></li><li><a href="/wiki/Category:Cleanup_tagged_articles_with_a_reason_field_from_May_2024" title="Category:Cleanup tagged articles with a reason field from May 2024">Cleanup tagged articles with a reason field from May 2024</a></li><li><a href="/wiki/Category:Wikipedia_pages_needing_cleanup_from_May_2024" title="Category:Wikipedia pages needing cleanup from May 2024">Wikipedia pages needing cleanup from May 2024</a></li><li><a href="/wiki/Category:Pages_using_multiple_image_with_auto_scaled_images" title="Category:Pages using multiple image with auto scaled images">Pages using multiple image with auto scaled images</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 November 2024, at 20:27 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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