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<span class="vector-toc-numb">2</span> <span>Variations and uses</span> </div> </a> <ul id="toc-Variations_and_uses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Star_polygons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Star_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Star polygons</span> </div> </a> <ul id="toc-Star_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverted_polygons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inverted_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Inverted polygons</span> </div> </a> <ul id="toc-Inverted_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-All_uniform_vertex_configurations_of_regular_convex_polygons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#All_uniform_vertex_configurations_of_regular_convex_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>All uniform vertex configurations of regular convex polygons</span> </div> </a> <ul id="toc-All_uniform_vertex_configurations_of_regular_convex_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Face_configuration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Face_configuration"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Face configuration</span> </div> </a> <ul id="toc-Face_configuration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> 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configuration</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://es.wikipedia.org/wiki/Configuraci%C3%B3n_de_v%C3%A9rtices" title="Configuración de vértices – Spanish" lang="es" hreflang="es" data-title="Configuración de vértices" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vertica_konfiguro" title="Vertica konfiguro – Esperanto" lang="eo" hreflang="eo" data-title="Vertica konfiguro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Configuration_de_sommet" title="Configuration de sommet – French" lang="fr" hreflang="fr" data-title="Configuration de sommet" data-language-autonym="Français" data-language-local-name="French" 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configuration</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Notation for a polyhedron's vertex figure</div> <table class="wikitable" align="right"> <tbody><tr valign="top" align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Icosidodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/200px-Icosidodecahedron.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/300px-Icosidodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/400px-Icosidodecahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Icosidodecahedron_vertfig_labeled.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Icosidodecahedron_vertfig_labeled.png/150px-Icosidodecahedron_vertfig_labeled.png" decoding="async" width="150" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Icosidodecahedron_vertfig_labeled.png/225px-Icosidodecahedron_vertfig_labeled.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Icosidodecahedron_vertfig_labeled.png/300px-Icosidodecahedron_vertfig_labeled.png 2x" data-file-width="374" data-file-height="518" /></a></span><br /><a href="/wiki/Vertex_figure" title="Vertex figure">Vertex figure</a> represented as<br /><span class="texhtml">3.5.3.5</span> or <span class="texhtml">(3.5)<sub>2</sub></span> </td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>vertex configuration</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Steurer_3-0" class="reference"><a href="#cite_note-Steurer-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Laughlin_4-0" class="reference"><a href="#cite_note-Laughlin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> is a shorthand notation for representing the <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="According to the linked article, "a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off." You can use a vertex configuration to represent a polyhedron or tiling even when there is no corner that has been cut off. This becomes especially obvious when you use a vertex configuration (such as 3.6.3.6) to represent a tiling of a plane, which can impossibly ever have had any corner that has been cut off. (August 2024)">dubious</span></a> – <a href="/wiki/Talk:Vertex_configuration#Dubious" title="Talk:Vertex configuration">discuss</a></i>]</sup> of a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> or <a href="/wiki/Tessellation" title="Tessellation">tiling</a> as the sequence of <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> around a <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a>. For <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform polyhedra</a> there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (<a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">Chiral</a> polyhedra exist in mirror-image pairs with the same vertex configuration.) </p><p>A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "<span class="texhtml mvar" style="font-style:italic;">a.b.c</span>" describes a vertex that has 3 faces around it, faces with <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> sides. </p><p>For example, "<span class="texhtml">3.5.3.5</span>" indicates a vertex belonging to 4 faces, alternating <a href="/wiki/Triangle" title="Triangle">triangles</a> and <a href="/wiki/Pentagon" title="Pentagon">pentagons</a>. This vertex configuration defines the <a href="/wiki/Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a> <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a>. The notation is cyclic and therefore is equivalent with different starting points, so <span class="texhtml">3.5.3.5</span> is the same as <span class="texhtml">5.3.5.3.</span> The order is important, so <span class="texhtml">3.3.5.5</span> is different from <span class="texhtml">3.5.3.5</span> (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as <span class="texhtml">(3.5)<sub>2</sub></span>. </p><p>It has variously been called a <b>vertex description</b>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <b>vertex type</b>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <b>vertex symbol</b>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <b>vertex arrangement</b>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <b>vertex pattern</b>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <b>face-vector</b>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> It is also called a <b><a href="/wiki/Martyn_Cundy" title="Martyn Cundy">Cundy</a> and Rollett symbol</b> for its usage for the <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a> in their 1952 book <i><a href="/wiki/Mathematical_Models_(Cundy_and_Rollett)" title="Mathematical Models (Cundy and Rollett)">Mathematical Models</a></i>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Vertex_figures">Vertex figures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=1" title="Edit section: Vertex figures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>vertex configuration</i> can also be represented as a <a href="/wiki/Polygon" title="Polygon">polygonal</a> <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> showing the faces around the vertex. This <i>vertex figure</i> has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">vertex-uniform polyhedra</a> all the neighboring vertices are in the same plane and so this <a href="/wiki/Orthographic_projection" title="Orthographic projection">plane projection</a> can be used to visually represent the vertex configuration. </p> <div class="mw-heading mw-heading2"><h2 id="Variations_and_uses">Variations and uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=2" title="Edit section: Variations and uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right" style="text-align:center;"> <caption>Regular vertex figure nets, {<i>p</i>,<i>q</i>} = <i>p<sup>q</sup></i> </caption> <tbody><tr valign="bottom"> <td><span typeof="mw:File"><a href="/wiki/File:Polyiamond-3-1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Polyiamond-3-1.svg/80px-Polyiamond-3-1.svg.png" decoding="async" width="80" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Polyiamond-3-1.svg/120px-Polyiamond-3-1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Polyiamond-3-1.svg/160px-Polyiamond-3-1.svg.png 2x" data-file-width="446" data-file-height="208" /></a></span><br /><a href="/wiki/Tetrahedron" title="Tetrahedron">{3,3}</a> = 3<sup>3</sup><br />Defect 180° </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyiamond-4-1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Polyiamond-4-1.svg/80px-Polyiamond-4-1.svg.png" decoding="async" width="80" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Polyiamond-4-1.svg/120px-Polyiamond-4-1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Polyiamond-4-1.svg/160px-Polyiamond-4-1.svg.png 2x" data-file-width="446" data-file-height="393" /></a></span><br /><a href="/wiki/Octahedron" title="Octahedron">{3,4}</a> = 3<sup>4</sup><br />Defect 120° </td> <td><span typeof="mw:File"><a href="/wiki/File:Polyiamond-5-4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Polyiamond-5-4.svg/80px-Polyiamond-5-4.svg.png" decoding="async" width="80" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Polyiamond-5-4.svg/120px-Polyiamond-5-4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Polyiamond-5-4.svg/160px-Polyiamond-5-4.svg.png 2x" data-file-width="446" data-file-height="393" /></a></span><br /><a href="/wiki/Icosahedron" title="Icosahedron">{3,5}</a> = 3<sup>5</sup><br />Defect 60° </td> <td bgcolor="#e0e0ff"><span typeof="mw:File"><a href="/wiki/File:Polyiamond-6-11.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Polyiamond-6-11.svg/80px-Polyiamond-6-11.svg.png" decoding="async" width="80" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Polyiamond-6-11.svg/120px-Polyiamond-6-11.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Polyiamond-6-11.svg/160px-Polyiamond-6-11.svg.png 2x" data-file-width="446" data-file-height="393" /></a></span><br /><a href="/wiki/Triangular_tiling" title="Triangular tiling">{3,6}</a> = <p>3<sup>6</sup><br />Defect 0° </p> </td></tr> <tr valign="bottom"> <td><span typeof="mw:File"><a href="/wiki/File:TrominoV.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/TrominoV.jpg/80px-TrominoV.jpg" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/TrominoV.jpg/120px-TrominoV.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/TrominoV.jpg/160px-TrominoV.jpg 2x" data-file-width="200" data-file-height="200" /></a></span><br /><a href="/wiki/Cube" title="Cube">{4,3}</a><br />Defect 90° </td> <td bgcolor="#e0e0ff"><span typeof="mw:File"><a href="/wiki/File:Square_tiling_vertfig.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Square_tiling_vertfig.svg/80px-Square_tiling_vertfig.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Square_tiling_vertfig.svg/120px-Square_tiling_vertfig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Square_tiling_vertfig.svg/160px-Square_tiling_vertfig.svg.png 2x" data-file-width="612" data-file-height="612" /></a></span><br /><a href="/wiki/Square_tiling" title="Square tiling">{4,4}</a> = <p>4<sup>4</sup><br />Defect 0° </p> </td> <td><span typeof="mw:File"><a href="/wiki/File:Pentagon_net.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Pentagon_net.png/80px-Pentagon_net.png" decoding="async" width="80" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Pentagon_net.png/120px-Pentagon_net.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Pentagon_net.png/160px-Pentagon_net.png 2x" data-file-width="1913" data-file-height="1526" /></a></span><br /><a href="/wiki/Dodecahedron" title="Dodecahedron">{5,3}</a> = 5<sup>3</sup><br />Defect 36° </td> <td bgcolor="#e0e0ff"><span typeof="mw:File"><a href="/wiki/File:Hexagonal_tiling_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Hexagonal_tiling_vertfig.png/80px-Hexagonal_tiling_vertfig.png" decoding="async" width="80" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Hexagonal_tiling_vertfig.png/120px-Hexagonal_tiling_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Hexagonal_tiling_vertfig.png/160px-Hexagonal_tiling_vertfig.png 2x" data-file-width="536" data-file-height="531" /></a></span><br /><a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">{6,3}</a> = <p>6<sup>3</sup><br />Defect 0° </p> </td></tr> <tr> <td colspan="4">A vertex needs at least 3 faces, and an <a href="/wiki/Angle_defect" class="mw-redirect" title="Angle defect">angle defect</a>.<br />A 0° angle defect will fill the Euclidean plane with a regular tiling.<br />By <a href="/wiki/Angular_defect#Descartes.27_theorem" title="Angular defect">Descartes' theorem</a>, the number of vertices is 720°/<i>defect</i> (4π radians/<i>defect</i>). </td></tr></tbody></table> <p>Different notations are used, sometimes with a comma (,) and sometimes a period (.) separator. The period operator is useful because it looks like a product and an exponent notation can be used. For example, 3.5.3.5 is sometimes written as (3.5)<sup>2</sup>. </p><p>The notation can also be considered an expansive form of the simple <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> for <a href="/wiki/Platonic_solid" title="Platonic solid">regular polyhedra</a>. The Schläfli notation {<i>p</i>,<i>q</i>} means <i>q</i> <i>p</i>-gons around each vertex. So {<i>p</i>,<i>q</i>} can be written as <i>p.p.p...</i> (<i>q</i> times) or <i>p<sup>q</sup></i>. For example, an icosahedron is {3,5} = 3.3.3.3.3 or 3<sup>5</sup>. </p><p>This notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron. </p><p>The notation is ambiguous for <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a> forms. For example, the <a href="/wiki/Snub_cube" title="Snub cube">snub cube</a> has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration. </p> <div class="mw-heading mw-heading2"><h2 id="Star_polygons">Star polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=3" title="Edit section: Star polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The notation also applies for nonconvex regular faces, the <a href="/wiki/Star_polygon" title="Star polygon">star polygons</a>. For example, a <a href="/wiki/Pentagram" title="Pentagram">pentagram</a> has the symbol {5/2}, meaning it has 5 sides going around the centre twice. </p><p>For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The <a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">small stellated dodecahedron</a> has the <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> of {5/2,5} which expands to an explicit vertex configuration 5/2.5/2.5/2.5/2.5/2 or combined as (5/2)<sup>5</sup>. The <a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">great stellated dodecahedron</a>, {5/2,3} has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2)<sup>3</sup>. The <a href="/wiki/Great_dodecahedron" title="Great dodecahedron">great dodecahedron</a>, {5,5/2} has a pentagrammic vertex figure, with <i>vertex configuration</i> is (5.5.5.5.5)/2 or (5<sup>5</sup>)/2. A <a href="/wiki/Great_icosahedron" title="Great icosahedron">great icosahedron</a>, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration (3.3.3.3.3)/2 or (3<sup>5</sup>)/2. </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Small_stellated_dodecahedron_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Small_stellated_dodecahedron_vertfig.png/150px-Small_stellated_dodecahedron_vertfig.png" decoding="async" width="150" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Small_stellated_dodecahedron_vertfig.png/225px-Small_stellated_dodecahedron_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Small_stellated_dodecahedron_vertfig.png/300px-Small_stellated_dodecahedron_vertfig.png 2x" data-file-width="658" data-file-height="627" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:Great_stellated_dodecahedron_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Great_stellated_dodecahedron_vertfig.png/150px-Great_stellated_dodecahedron_vertfig.png" decoding="async" width="150" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Great_stellated_dodecahedron_vertfig.png/225px-Great_stellated_dodecahedron_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Great_stellated_dodecahedron_vertfig.png/300px-Great_stellated_dodecahedron_vertfig.png 2x" data-file-width="495" data-file-height="538" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Great_snub_icosidodecahedron_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Great_snub_icosidodecahedron_vertfig.png/150px-Great_snub_icosidodecahedron_vertfig.png" decoding="async" width="150" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Great_snub_icosidodecahedron_vertfig.png/225px-Great_snub_icosidodecahedron_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Great_snub_icosidodecahedron_vertfig.png/300px-Great_snub_icosidodecahedron_vertfig.png 2x" data-file-width="579" data-file-height="561" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Great_retrosnub_icosidodecahedron_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Great_retrosnub_icosidodecahedron_vertfig.png/150px-Great_retrosnub_icosidodecahedron_vertfig.png" decoding="async" width="150" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Great_retrosnub_icosidodecahedron_vertfig.png/225px-Great_retrosnub_icosidodecahedron_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Great_retrosnub_icosidodecahedron_vertfig.png/300px-Great_retrosnub_icosidodecahedron_vertfig.png 2x" data-file-width="499" data-file-height="502" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Small_retrosnub_icosicosidodecahedron_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Small_retrosnub_icosicosidodecahedron_vertfig.png/150px-Small_retrosnub_icosicosidodecahedron_vertfig.png" decoding="async" width="150" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Small_retrosnub_icosicosidodecahedron_vertfig.png/225px-Small_retrosnub_icosicosidodecahedron_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Small_retrosnub_icosicosidodecahedron_vertfig.png/300px-Small_retrosnub_icosicosidodecahedron_vertfig.png 2x" data-file-width="453" data-file-height="549" /></a></span> </td></tr> <tr> <th><a href="/wiki/Small_stellated_dodecahedron" title="Small stellated dodecahedron">{5/2,5}</a> = (5/2)<sup>5</sup> </th> <th><a href="/wiki/Great_stellated_dodecahedron" title="Great stellated dodecahedron">{5/2,3}</a> = (5/2)<sup>3</sup> </th> <th><a href="/wiki/Great_snub_icosidodecahedron" title="Great snub icosidodecahedron">3<sup>4</sup>.5/2</a> </th> <th><a href="/wiki/Small_retrosnub_icosicosidodecahedron" title="Small retrosnub icosicosidodecahedron">3<sup>4</sup>.5/3</a> </th> <th><a href="/wiki/Great_retrosnub_icosidodecahedron" title="Great retrosnub icosidodecahedron">(3<sup>4</sup>.5/2)/2</a> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Great_dodecahedron_vertfig.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Great_dodecahedron_vertfig.png/150px-Great_dodecahedron_vertfig.png" decoding="async" width="150" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Great_dodecahedron_vertfig.png/225px-Great_dodecahedron_vertfig.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Great_dodecahedron_vertfig.png/300px-Great_dodecahedron_vertfig.png 2x" data-file-width="567" data-file-height="582" /></a></span></td> <td><span typeof="mw:File"><a href="/wiki/File:Great_icosahedron_vertfig.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Great_icosahedron_vertfig.svg/150px-Great_icosahedron_vertfig.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Great_icosahedron_vertfig.svg/225px-Great_icosahedron_vertfig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Great_icosahedron_vertfig.svg/300px-Great_icosahedron_vertfig.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:DU57_facets.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/DU57_facets.png/150px-DU57_facets.png" decoding="async" width="150" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/DU57_facets.png/225px-DU57_facets.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/DU57_facets.png/300px-DU57_facets.png 2x" data-file-width="1117" data-file-height="1173" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:DU72_facets.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/DU72_facets.png/150px-DU72_facets.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/DU72_facets.png/225px-DU72_facets.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/DU72_facets.png/300px-DU72_facets.png 2x" data-file-width="1200" data-file-height="1200" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:DU74_facets.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/DU74_facets.png/150px-DU74_facets.png" decoding="async" width="150" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/DU74_facets.png/225px-DU74_facets.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/DU74_facets.png/300px-DU74_facets.png 2x" data-file-width="1203" data-file-height="457" /></a></span> </td></tr> <tr> <th><a href="/wiki/Great_dodecahedron" title="Great dodecahedron">{5,5/2}</a> = (5<sup>5</sup>)/2 </th> <th><a href="/wiki/Great_icosahedron" title="Great icosahedron">{3,5/2}</a> = (3<sup>5</sup>)/2 </th> <th><a href="/wiki/Great_pentagonal_hexecontahedron" class="mw-redirect" title="Great pentagonal hexecontahedron">V.3<sup>4</sup>.5/2</a> </th> <th><a href="/wiki/Small_hexagrammic_hexecontahedron" title="Small hexagrammic hexecontahedron">V3<sup>4</sup>.5/3</a> </th> <th><a href="/wiki/Great_pentagrammic_hexecontahedron" title="Great pentagrammic hexecontahedron">V(3<sup>4</sup>.5/2)/2</a> </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Inverted_polygons">Inverted polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=4" title="Edit section: Inverted polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in the <a href="/wiki/Star_polygon" title="Star polygon">star polygon</a> notation of sides <i>p/q</i> such that <i>p</i><2<i>q</i>, where <i>p</i> is the number of sides and <i>q</i> the number of turns around a circle. For example, "3/2" means a triangle that has vertices that go around twice, which is the same as backwards once. Similarly "5/3" is a backwards pentagram 5/2. </p> <div class="mw-heading mw-heading2"><h2 id="All_uniform_vertex_configurations_of_regular_convex_polygons">All uniform vertex configurations of regular convex polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=5" title="Edit section: All uniform vertex configurations of regular convex polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Archimedean_solid#Classification" title="Archimedean solid">Archimedean_solid § Classification</a>, <a href="/wiki/Tiling_by_regular_polygons#Combinations_of_regular_polygons_that_can_meet_at_a_vertex" class="mw-redirect" title="Tiling by regular polygons">Tiling by regular polygons § Combinations of regular polygons that can meet at a vertex</a>, and <a href="/wiki/Uniform_tiling#Expanded_lists_of_uniform_tilings" title="Uniform tiling">Uniform tiling § Expanded lists of uniform tilings</a></div> <p><a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">Semiregular polyhedra</a> have vertex configurations with positive <a href="/wiki/Defect_(geometry)" class="mw-redirect" title="Defect (geometry)">angle defect</a>. </p><p>NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if its defect is zero. It can represent a tiling of the hyperbolic plane if its defect is negative. </p><p>For uniform polyhedra, the angle defect can be used to compute the number of vertices. Descartes' theorem states that all the angle defects in a topological sphere must sum to 4<i>π</i> radians or 720 degrees. </p><p>Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices, which is 4<i>π</i>/<i>defect</i> or 720/<i>defect</i>. </p><p>Example: A <a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a> 3.8.8 has an angle defect of 30 degrees. Therefore, it has <span class="texhtml">720/30 = 24</span> vertices. </p><p>In particular it follows that {<i>a</i>,<i>b</i>} has <span class="texhtml"> 4 / (2 - <i>b</i>(1 - 2/<i>a</i>))</span> vertices. </p><p>Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible. </p><p>Topological requirements limit existence. Specifically <i>p.q.r</i> implies that a <i>p</i>-gon is surrounded by alternating <i>q</i>-gons and <i>r</i>-gons, so either <i>p</i> is even or <i>q</i> equals <i>r</i>. Similarly <i>q</i> is even or <i>p</i> equals <i>r</i>, and <i>r</i> is even or <i>p</i> equals <i>q</i>. Therefore, potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.<i>n</i> (for any <i>n</i>>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6. In fact, all these configurations with three faces meeting at each vertex turn out to exist. </p><p>The number in parentheses is the number of vertices, determined by the angle defect. </p> <dl><dt>Triples</dt></dl> <ul><li>Platonic solids <a href="/wiki/Tetrahedron" title="Tetrahedron">3.3.3</a> (4), <a href="/wiki/Cube" title="Cube">4.4.4</a> (8), <a href="/wiki/Dodecahedron" title="Dodecahedron">5.5.5</a> (20)</li> <li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a> 3.4.4 (6), 4.4.4 (8; also listed above), 4.4.<i>n</i> (2<i>n</i>)</li> <li>Archimedean solids <a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">3.6.6</a> (12), <a href="/wiki/Truncated_cube" title="Truncated cube">3.8.8</a> (24), <a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">3.10.10</a> (60), <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">4.6.6</a> (24), <a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">4.6.8</a> (48), <a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">4.6.10</a> (120), <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">5.6.6</a> (60).</li> <li>regular tiling <a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">6.6.6</a></li> <li>semiregular tilings <a href="/wiki/Truncated_hexagonal_tiling" title="Truncated hexagonal tiling">3.12.12</a>, <a href="/wiki/Truncated_trihexagonal_tiling" title="Truncated trihexagonal tiling">4.6.12</a>, <a href="/wiki/Truncated_square_tiling" title="Truncated square tiling">4.8.8</a></li></ul> <dl><dt>Quadruples</dt></dl> <ul><li>Platonic solid <a href="/wiki/Octahedron" title="Octahedron">3.3.3.3</a> (6)</li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a> 3.3.3.3 (6; also listed above), 3.3.3.<i>n</i> (2<i>n</i>)</li> <li>Archimedean solids <a href="/wiki/Cuboctahedron" title="Cuboctahedron">3.4.3.4</a> (12), <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">3.5.3.5</a> (30), <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">3.4.4.4</a> (24), <a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">3.4.5.4</a> (60)</li> <li>regular tiling <a href="/wiki/Square_tiling" title="Square tiling">4.4.4.4</a></li> <li>semiregular tilings <a href="/wiki/Trihexagonal_tiling" title="Trihexagonal tiling">3.6.3.6</a>, <a href="/wiki/Rhombitrihexagonal_tiling" title="Rhombitrihexagonal tiling">3.4.6.4</a></li></ul> <dl><dt>Quintuples</dt></dl> <ul><li>Platonic solid <a href="/wiki/Icosahedron" title="Icosahedron">3.3.3.3.3</a> (12)</li> <li>Archimedean solids <a href="/wiki/Snub_cube" title="Snub cube">3.3.3.3.4</a> (24), <a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">3.3.3.3.5</a> (60) (both <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chiral</a>)</li> <li>semiregular tilings <a href="/wiki/Snub_hexagonal_tiling" class="mw-redirect" title="Snub hexagonal tiling">3.3.3.3.6</a> (chiral), <a href="/wiki/Elongated_triangular_tiling" title="Elongated triangular tiling">3.3.3.4.4</a>, <a href="/wiki/Snub_square_tiling" title="Snub square tiling">3.3.4.3.4</a> (note that the two different orders of the same numbers give two different patterns)</li></ul> <dl><dt>Sextuples</dt></dl> <ul><li>regular tiling <a href="/wiki/Triangular_tiling" title="Triangular tiling">3.3.3.3.3.3</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Face_configuration">Face configuration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=6" title="Edit section: Face configuration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rhombic_dodecahedron_v3434.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Rhombic_dodecahedron_v3434.png/220px-Rhombic_dodecahedron_v3434.png" decoding="async" width="220" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c7/Rhombic_dodecahedron_v3434.png 1.5x" data-file-width="225" data-file-height="281" /></a><figcaption><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Rhombic dodecahedron</a> </figcaption></figure> <p>The uniform dual or <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a>, including the <a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a> and <a href="/wiki/Trapezohedra" class="mw-redirect" title="Trapezohedra">trapezohedra</a>, are <i>vertically-regular</i> (<a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>) and so they can be identified by a similar notation which is sometimes called <b>face configuration</b>.<sup id="cite_ref-Steurer_3-1" class="reference"><a href="#cite_note-Steurer-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Cundy and Rollett prefixed these dual symbols by a <i>V</i>. In contrast, <i><a href="/wiki/Tilings_and_patterns" title="Tilings and patterns">Tilings and patterns</a></i> uses square brackets around the symbol for isohedral tilings. </p><p>This notation represents a sequential count of the number of faces that exist at each <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> around a <a href="/wiki/Face_(geometry)" title="Face (geometry)">face</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> For example, V3.4.3.4 or V(3.4)<sup>2</sup> represents the <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> which is face-transitive: every face is a <a href="/wiki/Rhombus" title="Rhombus">rhombus</a>, and alternating vertices of the rhombus contain 3 or 4 faces each. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.math.technion.ac.il/S/rl/docs/uniform.pdf">Uniform Solution for Uniform Polyhedra</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151127053535/http://www.math.technion.ac.il/S/rl/docs/uniform.pdf">Archived</a> 2015-11-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (1993)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mathconsult.ch/static/unipoly/unipoly.html">The Uniform Polyhedra</a> Roman E. Maeder (1995)</span> </li> <li id="cite_note-Steurer-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Steurer_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Steurer_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=nVx-tu596twC&pg=PA18">Crystallography of Quasicrystals: Concepts, Methods and Structures</a> by Walter Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53</span> </li> <li id="cite_note-Laughlin-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Laughlin_4-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=jBZ0AwAAQBAJ&pg=PA20">Physical Metallurgy: 3-Volume Set, Volume 1</a> edited by David E. Laughlin, (2014) pp. 16–20</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.uwgb.edu/dutchs/symmetry/archpol.htm">Archimedean Polyhedra</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170705034044/https://www.uwgb.edu/dutchs/symmetry/archpol.htm">Archived</a> 2017-07-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Steven Dutch</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.orchidpalms.com/polyhedra/uniform/uniform.html">Uniform Polyhedra</a> Jim McNeill</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.software3d.com/Uniform.php">Uniform Polyhedra and their Duals</a> Robert Webb</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://hrcak.srce.hr/file/111177/">Symmetry-type graphs of Platonic and Archimedean solids</a>, Jurij Kovič, (2011)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://people.hws.edu/mitchell/tilings/part3.html">3. General Theorems: Regular and Semi-Regular Tilings</a> Kevin Mitchell, 1995</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Resources for Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://polyhedra.mathmos.net/entry/vertexsymbol.html">Vertex Symbol</a> Robert Whittaker</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Structure and Form in Design: Critical Ideas for Creative Practice By Michael Hann</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/264848030_Symmetry-type_graphs_of_Platonic_and_Archimedean_solids">Symmetry-type graphs of Platonic and Archimedean solids</a> Jurij Kovič</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDezaShtogrin2000" class="citation cs2">Deza, Michel; Shtogrin, Mikhail (2000), "Uniform partitions of 3-space, their relatives and embedding", <i>European Journal of Combinatorics</i>, <b>21</b> (6): 807–814, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9906034">math/9906034</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Feujc.1999.0385">10.1006/eujc.1999.0385</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1791208">1791208</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Combinatorics&rft.atitle=Uniform+partitions+of+3-space%2C+their+relatives+and+embedding&rft.volume=21&rft.issue=6&rft.pages=807-814&rft.date=2000&rft_id=info%3Aarxiv%2Fmath%2F9906034&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1791208%23id-name%3DMR&rft_id=info%3Adoi%2F10.1006%2Feujc.1999.0385&rft.aulast=Deza&rft.aufirst=Michel&rft.au=Shtogrin%2C+Mikhail&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+configuration" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Archimedean_solid"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><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/ArchimedeanSolid.html">"Archimedean solid"</a>, <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Archimedean+solid&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FArchimedeanSolid.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+configuration" class="Z3988"></span></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA164">Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere</a> 6.4.1 Cundy-Rollett symbol, p. 164</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Laughlin (2014), p. 16</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Cundy and Rollett (1952)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Martyn_Cundy" title="Martyn Cundy">Cundy, H.</a> and Rollett, A., <i><a href="/wiki/Mathematical_Models_(Cundy_and_Rollett)" title="Mathematical Models (Cundy and Rollett)">Mathematical Models</a></i> (1952), (3rd edition, 1989, Stradbroke, England: Tarquin Pub.), 3.7 <i>The Archimedean Polyhedra</i>. Pp. 101–115, pp. 118–119 Table I, Nets of Archimedean Duals, V.<i>a</i>.<i>b</i>.<i>c</i>... as <i>vertically-regular</i> symbols.</li> <li>Peter Cromwell, <i><a href="/wiki/Polyhedra_(book)" title="Polyhedra (book)">Polyhedra</a></i>, Cambridge University Press (1977) The Archimedean solids. Pp. 156–167.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams1979" class="citation book cs1"><a href="/wiki/Robert_Williams_(geometer)" title="Robert Williams (geometer)">Williams, Robert</a> (1979). <i>The Geometrical Foundation of Natural Structure: A Source Book of Design</i>. Dover Publications, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-23729-X" title="Special:BookSources/0-486-23729-X"><bdi>0-486-23729-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometrical+Foundation+of+Natural+Structure%3A+A+Source+Book+of+Design&rft.pub=Dover+Publications%2C+Inc&rft.date=1979&rft.isbn=0-486-23729-X&rft.aulast=Williams&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+configuration" class="Z3988"></span> Uses Cundy-Rollett symbol.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrünbaum,_BrankoShephard,_G._C.1987" class="citation book cs1"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a>; <a href="/wiki/G.C._Shephard" class="mw-redirect" title="G.C. Shephard">Shephard, G. C.</a> (1987). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_0716711931"><i>Tilings and Patterns</i></a></span>. W. H. Freeman and Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-1193-1" title="Special:BookSources/0-7167-1193-1"><bdi>0-7167-1193-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tilings+and+Patterns&rft.pub=W.+H.+Freeman+and+Company&rft.date=1987&rft.isbn=0-7167-1193-1&rft.au=Gr%C3%BCnbaum%2C+Branko&rft.au=Shephard%2C+G.+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_0716711931&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertex+configuration" class="Z3988"></span> Pp. 58–64, Tilings of regular polygons a.b.c.... (Tilings by regular polygons and star polygons) pp. 95–97, 176, 283, 614–620, Monohedral tiling symbol [v<sub>1</sub>.v<sub>2</sub>. ... .v<sub>r</sub>]. pp. 632–642 hollow tilings.</li> <li><i>The Symmetries of Things</i> 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, <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> (p. 289 Vertex figures, uses comma separator, for Archimedean solids and tilings).</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertex_configuration&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.software3d.com/VertexDesc.php">Consistent Vertex Descriptions</a> <a href="/wiki/Stella_(software)" title="Stella (software)">Stella (software)</a>, Robert Webb</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=Vertex_configuration&oldid=1259618488#Face_configuration">https://en.wikipedia.org/w/index.php?title=Vertex_configuration&oldid=1259618488#Face_configuration</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">Categories</a>: <ul><li><a href="/wiki/Category:Polyhedra" title="Category:Polyhedra">Polyhedra</a></li><li><a href="/wiki/Category:Polytope_notation_systems" title="Category:Polytope notation systems">Polytope notation systems</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:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><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_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_accuracy_disputes" title="Category:All accuracy disputes">All accuracy disputes</a></li><li><a href="/wiki/Category:Articles_with_disputed_statements_from_August_2024" title="Category:Articles with disputed statements from August 2024">Articles with disputed statements from August 2024</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 26 November 2024, at 03:30<span class="anonymous-show"> (UTC)</span>.</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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