Conway polyhedron notation

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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Method of describing higher-order polyhedra</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Conway_relational_chart.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Conway_relational_chart.png/400px-Conway_relational_chart.png" decoding="async" width="400" height="271" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Conway_relational_chart.png/600px-Conway_relational_chart.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Conway_relational_chart.png/800px-Conway_relational_chart.png 2x" data-file-width="1595" data-file-height="1080" /></a><figcaption>This example chart shows how 11 new forms can be derived from the cube using 3 operations. The new polyhedra are shown as maps on the surface of the cube so the topological changes are more apparent. Vertices are marked in all forms with circles.</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Topology" title="Topology">topology</a>, <b>Conway polyhedron notation</b>, invented by <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a> and promoted by <a href="/wiki/George_W._Hart" title="George W. Hart">George W. Hart</a>, is used to describe <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a> based on a seed polyhedron modified by various prefix <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a>.<sup id="cite_ref-SoT_1-0" class="reference"><a href="#cite_note-SoT-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> </p><p>Conway and Hart extended the idea of using operators, like <a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">truncation</a> as defined by <a href="/wiki/Kepler" class="mw-redirect" title="Kepler">Kepler</a>, to build related polyhedra of the same symmetry. For example, <span class="texhtml mvar" style="font-style:italic;">tC</span> represents a <a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a>, and <span class="texhtml mvar" style="font-style:italic;">taC</span>, parsed as <span class="texhtml"><i>t</i>(<i>aC</i>)</span>, is (<a href="/wiki/Topology" title="Topology">topologically</a>) a <a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a>. The simplest operator <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> swaps <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertex</a> and <a href="/wiki/Face_(geometry)" title="Face (geometry)">face</a> elements; e.g., a dual <a href="/wiki/Cube" title="Cube">cube</a> is an <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>: <span class="texhtml"><i>dC</i> = <i>O</i></span>. Applied in a series, these operators allow many <a href="/w/index.php?title=Higher_order_polyhedra&action=edit&redlink=1" class="new" title="Higher order polyhedra (page does not exist)">higher order polyhedra</a> to be generated. Conway defined the operators <span class="texhtml mvar" style="font-style:italic;">a</span> (ambo), <span class="texhtml mvar" style="font-style:italic;">b</span> (<a href="/wiki/Bevel" title="Bevel">bevel</a>), <span class="texhtml mvar" style="font-style:italic;">d</span> (<a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a>), <span class="texhtml mvar" style="font-style:italic;">e</span> (expand), <span class="texhtml mvar" style="font-style:italic;">g</span> (gyro), <span class="texhtml mvar" style="font-style:italic;">j</span> (join), <span class="texhtml mvar" style="font-style:italic;">k</span> (kis), <span class="texhtml mvar" style="font-style:italic;">m</span> (meta), <span class="texhtml mvar" style="font-style:italic;">o</span> (ortho), <span class="texhtml mvar" style="font-style:italic;">s</span> (<a href="/wiki/Snub_(geometry)" title="Snub (geometry)">snub</a>), and <span class="texhtml mvar" style="font-style:italic;">t</span> (<a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">truncate</a>), while Hart added <span class="texhtml mvar" style="font-style:italic;">r</span> (<a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflect</a>) and <span class="texhtml mvar" style="font-style:italic;">p</span> (propellor).<sup id="cite_ref-Hart_3-0" class="reference"><a href="#cite_note-Hart-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Later implementations named further operators, sometimes referred to as "extended" operators.<sup id="cite_ref-Antiprism_4-0" class="reference"><a href="#cite_note-Antiprism-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Levskaya_5-0" class="reference"><a href="#cite_note-Levskaya-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Conway's basic operations are sufficient to generate the <a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean</a> and <a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a> from the <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a>. Some basic operations can be made as composites of others: for instance, ambo applied twice is the expand operation (<span class="texhtml"><i>aa</i> = <i>e</i></span>), while a truncation after ambo produces bevel (<span class="texhtml"><i>ta</i> = <i>b</i></span>). </p><p>Polyhedra can be studied topologically, in terms of how their vertices, edges, and faces connect together, or geometrically, in terms of the placement of those elements in space. Different implementations of these operators may create polyhedra that are geometrically different but topologically equivalent. These topologically equivalent polyhedra can be thought of as one of many <a href="/wiki/Graph_embedding" title="Graph embedding">embeddings</a> of a <a href="/wiki/Polyhedral_graph" title="Polyhedral graph">polyhedral graph</a> on the sphere. Unless otherwise specified, in this article (and in the literature on Conway operators in general) topology is the primary concern. Polyhedra with <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> 0 (i.e. topologically equivalent to a sphere) are often put into <a href="/wiki/Canonical_polyhedron" class="mw-redirect" title="Canonical polyhedron">canonical form</a> to avoid ambiguity. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Operators">Operators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=1" title="Edit section: Operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Conway's notation, operations on polyhedra are applied like functions, from right to left. For example, a <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a> is an <i>ambo cube</i>,<sup id="cite_ref-hart_6-0" class="reference"><a href="#cite_note-hart-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> i.e. <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(C)=aC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(C)=aC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a831b3a2760d0c9246b6bf363f36d93b17a9599c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.9ex; height:2.843ex;" alt="{\displaystyle a(C)=aC}" /></span>⁠</span>, and a <a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a> is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(a(C))=t(aC)=taC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mi>a</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(a(C))=t(aC)=taC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8faca97e2ef0d1fac59c951b6bca5b4bc11dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.132ex; height:2.843ex;" alt="{\displaystyle t(a(C))=t(aC)=taC}" /></span>⁠</span>. Repeated application of an operator can be denoted with an exponent: <i>j<sup>2</sup></i> = <i>o</i>. In general, Conway operators are not <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>. </p><p>Individual operators can be visualized in terms of <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domains</a> (or chambers), as below. Each right triangle is a <a href="/wiki/Fundamental_domain" title="Fundamental domain">fundamental domain</a>. Each white chamber is a rotated version of the others, and so is each colored chamber. For <a href="/wiki/Achiral" class="mw-redirect" title="Achiral">achiral</a> operators, the colored chambers are a reflection of the white chambers, and all are transitive. In group terms, achiral operators correspond to <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral groups</a> <span class="texhtml">D<sub><i>n</i></sub></span> where <i>n</i> is the number of sides of a face, while chiral operators correspond to <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> <span class="texhtml">C<sub><i>n</i></sub></span> lacking the reflective symmetry of the dihedral groups. Achiral and <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> operators are also called local symmetry-preserving operations (LSP) and local operations that preserve orientation-preserving symmetries (LOPSP), respectively.<sup id="cite_ref-Brinkmann_7-0" class="reference"><a href="#cite_note-Brinkmann-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lsp_8-0" class="reference"><a href="#cite_note-lsp-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-lopsp_9-0" class="reference"><a href="#cite_note-lopsp-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> LSPs should be understood as local operations that preserve symmetry, not operations that preserve local symmetry. Again, these are symmetries in a topological sense, not a geometric sense: the exact angles and edge lengths may differ. </p> <table class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"> <caption>Fundamental domains of faces with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> sides </caption> <tbody><tr> <th>3 (Triangle) </th> <th>4 (Square) </th> <th>5 (Pentagon) </th> <th>6 (Hexagon) </th></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Triangle_chambers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle_chambers.svg/183px-Triangle_chambers.svg.png" decoding="async" width="183" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle_chambers.svg/275px-Triangle_chambers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle_chambers.svg/366px-Triangle_chambers.svg.png 2x" data-file-width="200" data-file-height="176" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Quadrilateral_chambers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Quadrilateral_chambers.svg/170px-Quadrilateral_chambers.svg.png" decoding="async" width="170" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Quadrilateral_chambers.svg/255px-Quadrilateral_chambers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Quadrilateral_chambers.svg/340px-Quadrilateral_chambers.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Pentagon_chambers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Pentagon_chambers.svg/220px-Pentagon_chambers.svg.png" decoding="async" width="220" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Pentagon_chambers.svg/330px-Pentagon_chambers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Pentagon_chambers.svg/440px-Pentagon_chambers.svg.png 2x" data-file-width="200" data-file-height="191" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hexagon_chambers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Hexagon_chambers.svg/250px-Hexagon_chambers.svg.png" decoding="async" width="250" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Hexagon_chambers.svg/375px-Hexagon_chambers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Hexagon_chambers.svg/500px-Hexagon_chambers.svg.png 2x" data-file-width="200" data-file-height="176" /></a></span> </td></tr> <tr align="center"> <td colspan="4">The fundamental domains for polyhedron groups. The groups are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{3},D_{4},D_{5},D_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{3},D_{4},D_{5},D_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca5dcbf16e9b5d64b7a5f946ac24a75b49858e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.016ex; height:2.509ex;" alt="{\displaystyle D_{3},D_{4},D_{5},D_{6}}" /></span> for achiral polyhedra, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{3},C_{4},C_{5},C_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3},C_{4},C_{5},C_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b8f1ddbd3a3bc0c89e944f454a27c1c64f190c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.966ex; height:2.509ex;" alt="{\displaystyle C_{3},C_{4},C_{5},C_{6}}" /></span> for chiral polyhedra. </td></tr></tbody></table> <p>Hart introduced the reflection operator <i>r</i>, that gives the mirror image of the polyhedron.<sup id="cite_ref-hart_6-1" class="reference"><a href="#cite_note-hart-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This is not strictly a LOPSP, since it does not preserve orientation: it reverses it, by exchanging white and red chambers. <i>r</i> has no effect on achiral polyhedra aside from orientation, and <i>rr = S</i> returns the original polyhedron. An overline can be used to indicate the other chiral form of an operator: <span style="text-decoration:overline;"><i>s</i></span> = <i>rsr</i>. </p><p>An operation is irreducible if it cannot be expressed as a composition of operators aside from <i>d</i> and <i>r</i>. The majority of Conway's original operators are irreducible: the exceptions are <i>e</i>, <i>b</i>, <i>o</i>, and <i>m</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_representation">Matrix representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=2" title="Edit section: Matrix representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable floatright"> <tbody><tr> <th>x </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a&b&c\\0&d&0\\a'&b'&c'\end{bmatrix}}=\mathbf {M} _{x}}"> <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> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&b&c\\0&d&0\\a'&b'&c'\end{bmatrix}}=\mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1b59679b3474ac62ebf51cb7fe0a067a2caa31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.594ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}a&b&c\\0&d&0\\a'&b'&c'\end{bmatrix}}=\mathbf {M} _{x}}" /></span> </td></tr> <tr> <th>xd </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}c&b&a\\0&d&0\\c'&b'&a'\end{bmatrix}}=\mathbf {M} _{x}\mathbf {M} _{d}}"> <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> <mi>c</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>c</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}c&b&a\\0&d&0\\c'&b'&a'\end{bmatrix}}=\mathbf {M} _{x}\mathbf {M} _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662960908e3c96fa3b41730101a5f6c2e0430a49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.223ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}c&b&a\\0&d&0\\c'&b'&a'\end{bmatrix}}=\mathbf {M} _{x}\mathbf {M} _{d}}" /></span> </td></tr> <tr> <th>dx </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a'&b'&c'\\0&d&0\\a&b&c\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}}"> <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> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>c</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a'&b'&c'\\0&d&0\\a&b&c\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b703cca49916a460ffa227d8bf8141b05ff173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.223ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}a'&b'&c'\\0&d&0\\a&b&c\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}}" /></span> </td></tr> <tr> <th>dxd </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}c'&b'&a'\\0&d&0\\c&b&a\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}\mathbf {M} _{d}}"> <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> <msup> <mi>c</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> <mtd> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}c'&b'&a'\\0&d&0\\c&b&a\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}\mathbf {M} _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b8eaf11e9c50d49fad6907fa43d69422029c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.853ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}c'&b'&a'\\0&d&0\\c&b&a\end{bmatrix}}=\mathbf {M} _{d}\mathbf {M} _{x}\mathbf {M} _{d}}" /></span> </td></tr></tbody></table> <p>The relationship between the number of vertices, edges, and faces of the seed and the polyhedron created by the operations listed in this article can be expressed as a matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span>. When <i>x</i> is the operator, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,e,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,e,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7a363de90560e276c2e52402aceb2743c053b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.558ex; height:2.509ex;" alt="{\displaystyle v,e,f}" /></span> are the vertices, edges, and faces of the seed (respectively), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v',e',f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v',e',f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0dcca7ce5b8b74fe68231363452c7fe33c5661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.654ex; height:2.843ex;" alt="{\displaystyle v',e',f'}" /></span> are the vertices, edges, and faces of the result, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}{\begin{bmatrix}v\\e\\f\end{bmatrix}}={\begin{bmatrix}v'\\e'\\f'\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>v</mi> </mtd> </mtr> <mtr> <mtd> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>v</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}{\begin{bmatrix}v\\e\\f\end{bmatrix}}={\begin{bmatrix}v'\\e'\\f'\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26e72f6358a91133501d2a997163082eaa82a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.796ex; height:9.509ex;" alt="{\displaystyle \mathbf {M} _{x}{\begin{bmatrix}v\\e\\f\end{bmatrix}}={\begin{bmatrix}v'\\e'\\f'\end{bmatrix}}}" /></span>.</dd></dl> <p>The matrix for the composition of two operators is just the product of the matrixes for the two operators. Distinct operators may have the same matrix, for example, <i>p</i> and <i>l</i>. The edge count of the result is an integer multiple <i>d</i> of that of the seed: this is called the inflation rate, or the edge factor.<sup id="cite_ref-Brinkmann_7-1" class="reference"><a href="#cite_note-Brinkmann-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The simplest operators, the <a href="/wiki/Identity_operator" class="mw-redirect" title="Identity operator">identity operator</a> <i>S</i> and the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual operator</a> <i>d</i>, have simple matrix forms: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{S}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}=\mathbf {I} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{S}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}=\mathbf {I} _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3748296a540e1b4f90ff95863f049f2685af5ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.079ex; height:9.176ex;" alt="{\displaystyle \mathbf {M} _{S}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}=\mathbf {I} _{3}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{d}={\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{d}={\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f027ca30ecce9357173c689f8bba87b1cfff5036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.713ex; height:9.176ex;" alt="{\displaystyle \mathbf {M} _{d}={\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}}" /></span></dd></dl> <p>Two dual operators cancel out; <i>dd</i> = <i>S</i>, and the square of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04035b455e39f9a0d3e16453fb7eae5223b2c1bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.629ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{d}}" /></span> is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. When applied to other operators, the dual operator corresponds to horizontal and vertical reflections of the matrix. Operators can be grouped into groups of four (or fewer if some forms are the same) by identifying the operators <i>x</i>, <i>xd</i> (operator of dual), <i>dx</i> (dual of operator), and <i>dxd</i> (conjugate of operator). In this article, only the matrix for <i>x</i> is given, since the others are simple reflections. </p> <div class="mw-heading mw-heading3"><h3 id="Number_of_operators">Number of operators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=3" title="Edit section: Number of operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The number of LSPs for each inflation rate is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,2,4,6,6,20,28,58,82,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>28</mn> <mo>,</mo> <mn>58</mn> <mo>,</mo> <mn>82</mn> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,2,4,6,6,20,28,58,82,\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0f9e24bb9f5717adf2add6b58dfd2d84d4e3c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.141ex; height:2.509ex;" alt="{\displaystyle 2,2,4,6,6,20,28,58,82,\cdots }" /></span> starting with inflation rate 1. However, not all LSPs necessarily produce a polyhedron whose edges and vertices form a <a href="/wiki/K-vertex-connected_graph" title="K-vertex-connected graph">3-connected graph</a>, and as a consequence of <a href="/wiki/Steinitz%27s_theorem" title="Steinitz's theorem">Steinitz's theorem</a> do not necessarily produce a convex polyhedron from a convex seed. The number of 3-connected LSPs for each inflation rate is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,2,4,6,4,20,20,54,64,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>54</mn> <mo>,</mo> <mn>64</mn> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,2,4,6,4,20,20,54,64,\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1971b84d3e0cfe9932e9d774d47ad22f2688ac57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.141ex; height:2.509ex;" alt="{\displaystyle 2,2,4,6,4,20,20,54,64,\cdots }" /></span>.<sup id="cite_ref-lsp_8-1" class="reference"><a href="#cite_note-lsp-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Original_operations">Original operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=4" title="Edit section: Original operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Strictly, seed (<i>S</i>), needle (<i>n</i>), and zip (<i>z</i>) were not included by Conway, but they are related to original Conway operations by duality so are included here. </p><p>From here on, operations are visualized on cube seeds, drawn on the surface of that cube. Blue faces cross edges of the seed, and pink faces lie over vertices of the seed. There is some flexibility in the exact placement of vertices, especially with chiral operators. </p> <table class="wikitable sortable" style="text-align: center"> <caption>Original Conway operators </caption> <tbody><tr> <th>Edge factor</th> <th>Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span></th> <th>x</th> <th>xd</th> <th>dx</th> <th>dxd</th> <th>Notes </th></tr> <tr> <td>1 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917dc504a6780a695d578a7b216036af7e49c506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/100px-Conway_C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/150px-Conway_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/200px-Conway_C.png 2x" data-file-width="371" data-file-height="377" /></a></span><br /><b>Seed</b>: <i>S</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Conway_dC.png/100px-Conway_dC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Conway_dC.png/150px-Conway_dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6d/Conway_dC.png 2x" data-file-width="190" data-file-height="192" /></a></span><br /><b>Dual</b>: <i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/100px-Conway_C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/150px-Conway_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/200px-Conway_C.png 2x" data-file-width="371" data-file-height="377" /></a></span><br /><b>Seed</b>: <i>dd</i> = <i>S</i> </td> <td>Dual replaces each face with a vertex, and each vertex with a face. </td></tr> <tr> <td>2 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d1d55e81b3442e9a7775fa603c69f82dd3e9b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}}" /></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_jC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_jC.png/120px-Conway_jC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_jC.png/150px-Conway_jC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/51/Conway_jC.png 2x" data-file-width="184" data-file-height="192" /></a></span><br /><b>Join</b>: <i>j</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_aC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_aC.png/100px-Conway_aC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_aC.png/150px-Conway_aC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/87/Conway_aC.png 2x" data-file-width="195" data-file-height="198" /></a></span><br /><b>Ambo</b>: <i>a</i> </td> <td>Join creates quadrilateral faces. Ambo creates degree-4 vertices, and is also called <a href="/wiki/Rectification_(geometry)" title="Rectification (geometry)">rectification</a>, or the <a href="/wiki/Medial_graph" title="Medial graph">medial graph</a> in graph theory.<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> </td></tr> <tr> <td>3 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9419a43c4ffce9c1e51a5e52bee7c9a445ca72d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/120px-Conway_kC.png" decoding="async" width="100" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/150px-Conway_kC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b7/Conway_kC.png 2x" data-file-width="191" data-file-height="205" /></a></span><br /><b>Kis</b>: <i>k</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Conway_kdC.png/120px-Conway_kdC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Conway_kdC.png/250px-Conway_kdC.png 1.5x" data-file-width="372" data-file-height="377" /></a></span><br /><b>Needle</b>: <i>n</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dkC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/100px-Conway_dkC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/150px-Conway_dkC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/200px-Conway_dkC.png 2x" data-file-width="415" data-file-height="419" /></a></span><br /><b>Zip</b>: <i>z</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_tC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/100px-Conway_tC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/150px-Conway_tC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/200px-Conway_tC.png 2x" data-file-width="205" data-file-height="206" /></a></span><br /><b>Truncate</b>: <i>t</i> </td> <td>Kis raises a pyramid on each face, and is also called akisation, <a href="/wiki/Kleetope" title="Kleetope">Kleetope</a>, cumulation,<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> accretion, or pyramid-<a href="/wiki/Augmentation_(geometry)" class="mw-redirect" title="Augmentation (geometry)">augmentation</a>. <a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">Truncate</a> cuts off the polyhedron at its vertices but leaves a portion of the original edges.<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> Zip is also called <a href="/wiki/Bitruncation" title="Bitruncation">bitruncation</a>. </td></tr> <tr> <td>4 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0d385c1d46c4a5b8a4fb36c0abdec5f26a3800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_oC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Conway_oC.png/120px-Conway_oC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Conway_oC.png/150px-Conway_oC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/38/Conway_oC.png 2x" data-file-width="189" data-file-height="187" /></a></span><br /> <b>Ortho</b>: <i>o</i> = <i>jj</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_eC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_eC.png/100px-Conway_eC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_eC.png/150px-Conway_eC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/78/Conway_eC.png 2x" data-file-width="187" data-file-height="194" /></a></span><br /> <b>Expand</b>: <i>e</i> = <i>aa</i> </td> <td> </td></tr> <tr> <td>5 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1\\0&5&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1\\0&5&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ec126e243dcd542c4323812da7c5b9bdb71eed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1\\0&5&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_gC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_gC.png/100px-Conway_gC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_gC.png/150px-Conway_gC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_gC.png/200px-Conway_gC.png 2x" data-file-width="391" data-file-height="398" /></a></span><br /><b>Gyro</b>: <i>g</i> </td> <td><i>gd</i> = <i>rgr</i> </td> <td><i>sd</i> = <i>rsr</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_sC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Conway_sC.png/120px-Conway_sC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Conway_sC.png/250px-Conway_sC.png 1.5x" data-file-width="381" data-file-height="392" /></a></span><br /><b>Snub</b>: <i>s</i> </td> <td>Chiral operators. See <a href="/wiki/Snub_(geometry)" title="Snub (geometry)">Snub (geometry)</a>. Contrary to Hart,<sup id="cite_ref-Hart_3-1" class="reference"><a href="#cite_note-Hart-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <i>gd</i> is not the same as <i>g</i>: it is its chiral pair.<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> </td></tr> <tr> <td>6 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7605c575fe5a9100156fabebfadca202a26d0612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}}" /></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_mC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Conway_mC.png/100px-Conway_mC.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Conway_mC.png/150px-Conway_mC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0b/Conway_mC.png 2x" data-file-width="193" data-file-height="189" /></a></span><br /> <b>Meta</b>: <i>m</i> = <i>kj</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_bC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Conway_bC.png/100px-Conway_bC.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Conway_bC.png/150px-Conway_bC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7a/Conway_bC.png 2x" data-file-width="188" data-file-height="185" /></a></span><br /> <b>Bevel</b>: <i>b</i> = <i>ta</i> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Seeds">Seeds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=5" title="Edit section: Seeds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any polyhedron can serve as a seed, as long as the operations can be executed on it. Common seeds have been assigned a letter. The <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> are represented by the first letter of their name (<a href="/wiki/Tetrahedron" title="Tetrahedron"><b>T</b>etrahedron</a>, <a href="/wiki/Octahedron" title="Octahedron"><b>O</b>ctahedron</a>, <a href="/wiki/Cube" title="Cube"><b>C</b>ube</a>, <a href="/wiki/Icosahedron" title="Icosahedron"><b>I</b>cosahedron</a>, <a href="/wiki/Dodecahedron" title="Dodecahedron"><b>D</b>odecahedron</a>); the <a href="/wiki/Prism_(geometry)" title="Prism (geometry)"><b>p</b>risms</a> (<b>P</b><sub><i>n</i></sub>) for <i>n</i>-gonal forms; <a href="/wiki/Antiprism" title="Antiprism"><b>a</b>ntiprisms</a> (<b>A</b><sub><i>n</i></sub>); <a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">c<b>u</b>polae</a> (<b>U</b><sub><i>n</i></sub>); <a href="/wiki/Anticupola" class="mw-redirect" title="Anticupola">anticupolae</a> (<b>V</b><sub><i>n</i></sub>); and <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">p<b>y</b>ramids</a> (<b>Y</b><sub><i>n</i></sub>). Any <a href="/wiki/Johnson_solid" title="Johnson solid"><b>J</b>ohnson solid</a> can be referenced as <b>J</b><sub><i>n</i></sub>, for <i>n</i>=1..92. </p><p>All of the five Platonic solids can be generated from prismatic generators with zero to two operators:<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> </p> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><b><a href="/wiki/Triangular_pyramid" class="mw-redirect" title="Triangular pyramid">Triangular pyramid</a></b>: <i>Y</i><sub>3</sub> (A tetrahedron is a special pyramid) <ul><li><i><a href="/wiki/Tetrahedron" title="Tetrahedron">T</a></i> = <i>Y</i><sub>3</sub></li> <li><i><a href="/wiki/Octahedron" title="Octahedron">O</a></i> = <i>aT</i> (ambo tetrahedron)</li> <li><i><a href="/wiki/Cube" title="Cube">C</a></i> = <i>jT</i> (join tetrahedron)</li> <li><i><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">I</a></i> = <i>sT</i> (snub tetrahedron)</li> <li><i><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">D</a></i> = <i>gT</i> (gyro tetrahedron)</li></ul></li> <li><b><a href="/wiki/Antiprism" title="Antiprism">Triangular antiprism</a></b>: <i>A</i><sub>3</sub> (An octahedron is a special antiprism) <ul><li><i>O</i> = <i>A</i><sub>3</sub></li> <li><i>C</i> = <i>dA</i><sub>3</sub></li></ul></li> <li><b><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">Square prism</a></b>: <i>P</i><sub>4</sub> (A cube is a special prism) <ul><li><i>C</i> = <i>P</i><sub>4</sub></li></ul></li> <li><b><a href="/wiki/Pentagonal_antiprism" title="Pentagonal antiprism">Pentagonal antiprism</a></b>: <i>A</i><sub>5</sub> <ul><li><i>I</i> = <i>k</i><sub>5</sub><i>A</i><sub>5</sub> (A special <a href="/wiki/Gyroelongated_dipyramid" class="mw-redirect" title="Gyroelongated dipyramid">gyroelongated dipyramid</a>)</li> <li><i>D</i> = <i>t</i><sub>5</sub><i>dA</i><sub>5</sub> (A special <a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedron</a>)</li></ul></li></ul> </div> <p>The regular Euclidean tilings can also be used as seeds: </p> <ul><li><i>Q</i> = Quadrille = <a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a></li> <li><i>H</i> = Hextille = <a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">Hexagonal tiling</a> = <i>dΔ</i></li> <li><i>Δ</i> = Deltille = <a href="/wiki/Triangular_tiling" title="Triangular tiling">Triangular tiling</a> = <i>dH</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="Extended_operations">Extended operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=6" title="Edit section: Extended operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These are operations created after Conway's original set. Note that many more operations exist than have been named; just because an operation is not here does not mean it does not exist (or is not an LSP or LOPSP). In addition, only irreducible operators are included in this list; many others can be created by composing operators together. </p> <table class="wikitable sortable" style="text-align: center"> <caption>Irreducible extended operators </caption> <tbody><tr> <th>Edge factor</th> <th>Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span></th> <th>x</th> <th>xd</th> <th>dx</th> <th>dxd</th> <th>Notes </th></tr> <tr> <td>4 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&4&0\\0&1&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&4&0\\0&1&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8da1ee48cc9fa21cf08a99c509ea326f3d85a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&4&0\\0&1&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_cC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/100px-Conway_cC.png" decoding="async" width="100" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/150px-Conway_cC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/47/Conway_cC.png 2x" data-file-width="188" data-file-height="183" /></a></span><br /><b>Chamfer</b>: <i>c</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_duC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_duC.png/100px-Conway_duC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_duC.png/150px-Conway_duC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_duC.png/200px-Conway_duC.png 2x" data-file-width="382" data-file-height="389" /></a></span><br /><i>cd</i> = <i>du</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dcC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Conway_dcC.png/100px-Conway_dcC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Conway_dcC.png/150px-Conway_dcC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/70/Conway_dcC.png 2x" data-file-width="195" data-file-height="198" /></a></span><br /> <i>dc</i> = <i>ud</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_uC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_uC.png/120px-Conway_uC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_uC.png/250px-Conway_uC.png 1.5x" data-file-width="389" data-file-height="398" /></a></span><br /><b>Subdivide</b>: <i>u</i> </td> <td>Chamfer is the join-form of <i>l</i>. See <a href="/wiki/Chamfer_(geometry)" title="Chamfer (geometry)">Chamfer (geometry)</a>. </td></tr> <tr> <td>5 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17c726d82d518ea3355b897a9268fdaccd877df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_pC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/100px-Conway_pC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/150px-Conway_pC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/200px-Conway_pC.png 2x" data-file-width="381" data-file-height="392" /></a></span><br /><b>Propeller</b>: <i>p</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_dpC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Conway_dpC.png/100px-Conway_dpC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Conway_dpC.png/150px-Conway_dpC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Conway_dpC.png/200px-Conway_dpC.png 2x" data-file-width="375" data-file-height="380" /></a></span><br /><i>dp</i> = <i>pd</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_pC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/100px-Conway_pC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/150px-Conway_pC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/200px-Conway_pC.png 2x" data-file-width="381" data-file-height="392" /></a></span><br /><i>dpd</i> = <i>p</i> </td> <td>Chiral operators. The propeller operator was developed by George Hart.<sup id="cite_ref-Hart2000_15-0" class="reference"><a href="#cite_note-Hart2000-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>5 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17c726d82d518ea3355b897a9268fdaccd877df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_lC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Conway_lC.png/100px-Conway_lC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Conway_lC.png/150px-Conway_lC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Conway_lC.png/200px-Conway_lC.png 2x" data-file-width="379" data-file-height="382" /></a></span><br /><b>Loft</b>: <i>l</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_ldC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Conway_ldC.png/100px-Conway_ldC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Conway_ldC.png/150px-Conway_ldC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Conway_ldC.png/200px-Conway_ldC.png 2x" data-file-width="374" data-file-height="386" /></a></span><br /><i>ld</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dlC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Conway_dlC.png/100px-Conway_dlC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Conway_dlC.png/150px-Conway_dlC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Conway_dlC.png/200px-Conway_dlC.png 2x" data-file-width="379" data-file-height="383" /></a></span><br /> <i>dl</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dldC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Conway_dldC.png/120px-Conway_dldC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Conway_dldC.png/150px-Conway_dldC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Conway_dldC.png/200px-Conway_dldC.png 2x" data-file-width="203" data-file-height="204" /></a></span><br /><i>dld</i> </td> <td> </td></tr> <tr> <td>6 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&3&0\\0&6&0\\0&2&1\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> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&0\\0&6&0\\0&2&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4162bfce1e829c0b94ff59888f19ddd4b14a88ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&3&0\\0&6&0\\0&2&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_qC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Conway_qC.png/100px-Conway_qC.png" decoding="async" width="100" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Conway_qC.png/150px-Conway_qC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/de/Conway_qC.png 2x" data-file-width="193" data-file-height="202" /></a></span><br /><b>Quinto</b>: <i>q</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_qdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Conway_qdC.png/100px-Conway_qdC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Conway_qdC.png/150px-Conway_qdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/db/Conway_qdC.png 2x" data-file-width="199" data-file-height="207" /></a></span><br /><i>qd</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dqC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Conway_dqC.png/100px-Conway_dqC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Conway_dqC.png/150px-Conway_dqC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a5/Conway_dqC.png 2x" data-file-width="187" data-file-height="194" /></a></span><br /> <i>dq</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dqdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Conway_dqdC.png/100px-Conway_dqdC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Conway_dqdC.png/150px-Conway_dqdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Conway_dqdC.png/200px-Conway_dqdC.png 2x" data-file-width="389" data-file-height="401" /></a></span><br /><i>dqd</i> </td> <td> </td></tr> <tr> <td>6 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&6&0\\0&3&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&6&0\\0&3&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e965341c923b70ff7b167f2dbb44b59a1c77be35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&6&0\\0&3&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_L0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_L0C.png/100px-Conway_L0C.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_L0C.png/150px-Conway_L0C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_L0C.png/200px-Conway_L0C.png 2x" data-file-width="390" data-file-height="392" /></a></span><br /><b>Join-lace</b>: <i>L</i><sub>0</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_Diagram_L0d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Conway_Diagram_L0d.png/120px-Conway_Diagram_L0d.png" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Conway_Diagram_L0d.png/180px-Conway_Diagram_L0d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Conway_Diagram_L0d.png/240px-Conway_Diagram_L0d.png 2x" data-file-width="460" data-file-height="443" /></a></span><br /><i>L</i><sub>0</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dL0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_dL0C.png/100px-Conway_dL0C.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_dL0C.png/150px-Conway_dL0C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/50/Conway_dL0C.png 2x" data-file-width="191" data-file-height="197" /></a></span><br /><i>dL</i><sub>0</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dL0d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Conway_dL0d.png/100px-Conway_dL0d.png" decoding="async" width="100" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Conway_dL0d.png/150px-Conway_dL0d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Conway_dL0d.png/200px-Conway_dL0d.png 2x" data-file-width="401" data-file-height="389" /></a></span><br /><i>dL</i><sub>0</sub><i>d</i> </td> <td>See below for explanation of join notation. </td></tr> <tr> <td>7 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0b3ccc2dbba110409c7229b97e402ec7889fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_LC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conway_LC.png/100px-Conway_LC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conway_LC.png/150px-Conway_LC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conway_LC.png/200px-Conway_LC.png 2x" data-file-width="390" data-file-height="392" /></a></span><br /><b>Lace</b>: <i>L</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_L0dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Conway_L0dC.png/100px-Conway_L0dC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Conway_L0dC.png/150px-Conway_L0dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Conway_L0dC.png/200px-Conway_L0dC.png 2x" data-file-width="386" data-file-height="386" /></a></span><br /><i>Ld</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dLC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Conway_dLC.png/100px-Conway_dLC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Conway_dLC.png/150px-Conway_dLC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Conway_dLC.png/200px-Conway_dLC.png 2x" data-file-width="422" data-file-height="417" /></a></span><br /> <i>dL</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dLdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Conway_dLdC.png/100px-Conway_dLdC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Conway_dLdC.png/150px-Conway_dLdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Conway_dLdC.png/200px-Conway_dLdC.png 2x" data-file-width="401" data-file-height="400" /></a></span><br /><i>dLd</i> </td> <td> </td></tr> <tr> <td>7 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c1c28733dc91c447129dbd0798788f2dcbb4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_KC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Conway_KC.png/120px-Conway_KC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Conway_KC.png/250px-Conway_KC.png 1.5x" data-file-width="390" data-file-height="390" /></a></span><br /><b>Stake</b>: <i>K</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_KdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Conway_KdC.png/100px-Conway_KdC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Conway_KdC.png/150px-Conway_KdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Conway_KdC.png/200px-Conway_KdC.png 2x" data-file-width="392" data-file-height="396" /></a></span><br /><i>Kd</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dKC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Conway_dKC.png/100px-Conway_dKC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Conway_dKC.png/150px-Conway_dKC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Conway_dKC.png/200px-Conway_dKC.png 2x" data-file-width="499" data-file-height="493" /></a></span><br /> <i>dK</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dKdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Conway_dKdC.png/120px-Conway_dKdC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Conway_dKdC.png/250px-Conway_dKdC.png 1.5x" data-file-width="386" data-file-height="390" /></a></span><br /><i>dKd</i> </td> <td> </td></tr> <tr> <td>7 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&4&0\\0&7&0\\0&2&1\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> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&4&0\\0&7&0\\0&2&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a5d5ed1d66e172b0434ced012c74f88c7ef39f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&4&0\\0&7&0\\0&2&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_wC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Conway_wC.png/100px-Conway_wC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Conway_wC.png/150px-Conway_wC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Conway_wC.png/200px-Conway_wC.png 2x" data-file-width="397" data-file-height="398" /></a></span><br /><b>Whirl</b>: <i>w</i> </td> <td><i>wd</i> = <i>dv</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dwC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Conway_dwC.png/100px-Conway_dwC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Conway_dwC.png/150px-Conway_dwC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Conway_dwC.png/200px-Conway_dwC.png 2x" data-file-width="388" data-file-height="392" /></a></span><br /><i>vd</i> = <i>dw</i> </td> <td><b>Volute</b>: <i>v</i> </td> <td>Chiral operators. </td></tr> <tr> <td>8 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1\\0&8&0\\0&5&0\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1\\0&8&0\\0&5&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f13913bb870fae5775bd0674bf1fa96ac5e082d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1\\0&8&0\\0&5&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_(kk)0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Conway_%28kk%290C.png/100px-Conway_%28kk%290C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Conway_%28kk%290C.png/150px-Conway_%28kk%290C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Conway_%28kk%290C.png/200px-Conway_%28kk%290C.png 2x" data-file-width="375" data-file-height="381" /></a></span><br /><b>Join-kis-kis</b>: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (kk)_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mi>k</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (kk)_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31ca7ed4b1b42423c8a75c67fb5fdabdf15c3e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.286ex; height:2.843ex;" alt="{\displaystyle (kk)_{0}}" /></span>⁠</span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_(kk)0dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Conway_%28kk%290dC.png/100px-Conway_%28kk%290dC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Conway_%28kk%290dC.png/150px-Conway_%28kk%290dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Conway_%28kk%290dC.png/200px-Conway_%28kk%290dC.png 2x" data-file-width="387" data-file-height="385" /></a></span><br /> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (kk)_{0}d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mi>k</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (kk)_{0}d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04313c3aef185282f5637638633d21781079792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.502ex; height:2.843ex;" alt="{\displaystyle (kk)_{0}d}" /></span>⁠</span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_d(kk)0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Conway_d%28kk%290C.png/120px-Conway_d%28kk%290C.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Conway_d%28kk%290C.png/150px-Conway_d%28kk%290C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Conway_d%28kk%290C.png/200px-Conway_d%28kk%290C.png 2x" data-file-width="235" data-file-height="241" /></a></span><br /> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(kk)_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>k</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(kk)_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e48f7bd36769d12737da7c4def7cbbf171f1873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.502ex; height:2.843ex;" alt="{\displaystyle d(kk)_{0}}" /></span>⁠</span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_d(kk)0dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_d%28kk%290dC.png/100px-Conway_d%28kk%290dC.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_d%28kk%290dC.png/150px-Conway_d%28kk%290dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_d%28kk%290dC.png/200px-Conway_d%28kk%290dC.png 2x" data-file-width="237" data-file-height="233" /></a></span><br /> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(kk)_{0}d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>k</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(kk)_{0}d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb9e00aecaeb0ca692671e284b79d87a593e76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.718ex; height:2.843ex;" alt="{\displaystyle d(kk)_{0}d}" /></span>⁠</span> </td> <td>Sometimes named <i>J</i>.<sup id="cite_ref-Antiprism_4-1" class="reference"><a href="#cite_note-Antiprism-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> See below for explanation of join notation. The non-join-form, <i>kk</i>, is not irreducible. </td></tr> <tr> <td>10 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&3&1\\0&10&0\\0&6&0\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> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&1\\0&10&0\\0&6&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4228cce2d58d2b3be57393bf8cf4785622e1f490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&3&1\\0&10&0\\0&6&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_XC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Conway_XC.png/100px-Conway_XC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Conway_XC.png/150px-Conway_XC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Conway_XC.png/200px-Conway_XC.png 2x" data-file-width="389" data-file-height="398" /></a></span><br /><b>Cross</b>: <i>X</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_XdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Conway_XdC.png/100px-Conway_XdC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Conway_XdC.png/150px-Conway_XdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Conway_XdC.png/200px-Conway_XdC.png 2x" data-file-width="390" data-file-height="397" /></a></span><br /><i>Xd</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dXC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Conway_dXC.png/100px-Conway_dXC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Conway_dXC.png/150px-Conway_dXC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Conway_dXC.png/200px-Conway_dXC.png 2x" data-file-width="390" data-file-height="389" /></a></span><br /> <i>dX</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dXdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Conway_dXdC.png/100px-Conway_dXdC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Conway_dXdC.png/150px-Conway_dXdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Conway_dXdC.png/200px-Conway_dXdC.png 2x" data-file-width="383" data-file-height="389" /></a></span><br /><i>dXd</i> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Indexed_extended_operations">Indexed extended operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=7" title="Edit section: Indexed extended operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of operators can be grouped together by some criteria, or have their behavior modified by an index.<sup id="cite_ref-Antiprism_4-2" class="reference"><a href="#cite_note-Antiprism-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> These are written as an operator with a subscript: <i>x<sub>n</sub></i>. </p> <div class="mw-heading mw-heading3"><h3 id="Augmentation">Augmentation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=8" title="Edit section: Augmentation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Augmentation_(geometry)" class="mw-redirect" title="Augmentation (geometry)">Augmentation</a> operations retain original edges. They may be applied to any independent subset of faces, or may be converted into a <i>join</i>-form by removing the original edges. Conway notation supports an optional index to these operators: 0 for the join-form, or 3 or higher for how many sides affected faces have. For example, <i>k</i><sub>4</sub><i>Y</i><sub>4</sub>=O: taking a square-based pyramid and gluing another pyramid to the square base gives an octahedron. </p> <table class="wikitable" style="text-align: center"> <tbody><tr> <th>Augmentation operator </th> <td><i>x</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/120px-Conway_kC.png" decoding="async" width="80" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/160px-Conway_kC.png 2x" data-file-width="191" data-file-height="205" /></a></span> <p><i>k</i> </p> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_lC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Conway_lC.png/80px-Conway_lC.png" decoding="async" width="80" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Conway_lC.png/120px-Conway_lC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Conway_lC.png/160px-Conway_lC.png 2x" data-file-width="379" data-file-height="382" /></a></span> <p><i>l</i> </p> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_LC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conway_LC.png/80px-Conway_LC.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conway_LC.png/120px-Conway_LC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Conway_LC.png/160px-Conway_LC.png 2x" data-file-width="390" data-file-height="392" /></a></span> <p><i>L</i> </p> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_KC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Conway_KC.png/120px-Conway_KC.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Conway_KC.png/250px-Conway_KC.png 2x" data-file-width="390" data-file-height="390" /></a></span> <p><i>K</i> </p> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kkC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Conway_kkC.png/80px-Conway_kkC.png" decoding="async" width="80" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Conway_kkC.png/120px-Conway_kkC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Conway_kkC.png/160px-Conway_kkC.png 2x" data-file-width="405" data-file-height="410" /></a></span> <p><i>(kk)</i> </p> </td></tr> <tr> <th>Corresponding <p>join-form operator </p> </th> <td><i>x</i><sub>0</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_jC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_jC.png/120px-Conway_jC.png" decoding="async" width="80" height="83" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_jC.png/160px-Conway_jC.png 2x" data-file-width="184" data-file-height="192" /></a></span><br /><i>k</i><sub>0</sub> = <i>j</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_cC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/80px-Conway_cC.png" decoding="async" width="80" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/120px-Conway_cC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/160px-Conway_cC.png 2x" data-file-width="188" data-file-height="183" /></a></span><br /><i>l</i><sub>0</sub> = <i>c</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_L0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_L0C.png/80px-Conway_L0C.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_L0C.png/120px-Conway_L0C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_L0C.png/160px-Conway_L0C.png 2x" data-file-width="390" data-file-height="392" /></a></span><br /><i>L</i><sub>0</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_K0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_K0C.png/80px-Conway_K0C.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_K0C.png/120px-Conway_K0C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_K0C.png/160px-Conway_K0C.png 2x" data-file-width="390" data-file-height="390" /></a></span><br /><i>K</i><sub>0</sub> = <i>jk</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_(kk)0C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Conway_%28kk%290C.png/80px-Conway_%28kk%290C.png" decoding="async" width="80" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Conway_%28kk%290C.png/120px-Conway_%28kk%290C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Conway_%28kk%290C.png/160px-Conway_%28kk%290C.png 2x" data-file-width="375" data-file-height="381" /></a></span><br />(<i>kk</i>)<sub>0</sub> </td></tr> <tr align="center"> <th>Augmentation </th> <td> </td> <td><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></td> <td><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">Prism</a></td> <td><a href="/wiki/Antiprism" title="Antiprism">Antiprism</a></td> <td></td> <td> </td></tr></tbody></table> <p>The truncate operator <i>t</i> also has an index form <i>t<sub>n</sub></i>, indicating that only vertices of a certain degree are truncated. It is equivalent to <i>dk<sub>n</sub>d</i>. </p><p>Some of the extended operators can be created in special cases with <i>k<sub>n</sub></i> and <i>t<sub>n</sub></i> operators. For example, a <a href="/wiki/Chamfered_cube" class="mw-redirect" title="Chamfered cube">chamfered cube</a>, <i>cC</i>, can be constructed as <i>t</i><sub>4</sub><i>daC</i>, as a <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a>, <i>daC</i> or <i>jC</i>, with its degree-4 vertices truncated. A lofted cube, <i>lC</i> is the same as <i>t</i><sub>4</sub><i>kC</i>. A quinto-dodecahedron, <i>qD</i> can be constructed as <i>t</i><sub>5</sub><i>daaD</i> or <i>t</i><sub>5</sub><i>deD</i> or <i>t</i><sub>5</sub><i>oD</i>, a <a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a>, <i>deD</i> or <i>oD</i>, with its degree-5 vertices truncated. </p> <div class="mw-heading mw-heading3"><h3 id="Meta/Bevel"><span id="Meta.2FBevel"></span>Meta/Bevel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=9" title="Edit section: Meta/Bevel"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Meta adds vertices at the center and along the edges, while bevel adds faces at the center, seed vertices, and along the edges. The index is how many vertices or faces are added along the edges. Meta (in its non-indexed form) is also called <a href="/wiki/Cantitruncation" class="mw-redirect" title="Cantitruncation">cantitruncation</a> or <a href="/wiki/Omnitruncation" title="Omnitruncation">omnitruncation</a>. Note that 0 here does not mean the same as for augmentation operations: it means zero vertices (or faces) are added along the edges.<sup id="cite_ref-Antiprism_4-3" class="reference"><a href="#cite_note-Antiprism-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable sortable" style="text-align: center"> <caption>Meta/Bevel operators </caption> <tbody><tr> <th>n</th> <th>Edge factor</th> <th>Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span></th> <th>x</th> <th>xd</th> <th>dx</th> <th>dxd </th></tr> <tr> <td>0 </td> <td>3 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9419a43c4ffce9c1e51a5e52bee7c9a445ca72d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/120px-Conway_kC.png" decoding="async" width="100" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/150px-Conway_kC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b7/Conway_kC.png 2x" data-file-width="191" data-file-height="205" /></a></span><br /><i>k</i> = <i>m</i><sub>0</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Conway_kdC.png/120px-Conway_kdC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Conway_kdC.png/250px-Conway_kdC.png 1.5x" data-file-width="372" data-file-height="377" /></a></span><br /><i>n</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dkC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/100px-Conway_dkC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/150px-Conway_dkC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/200px-Conway_dkC.png 2x" data-file-width="415" data-file-height="419" /></a></span><br /><i>z</i> = <i>b</i><sub>0</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_tC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/100px-Conway_tC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/150px-Conway_tC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/200px-Conway_tC.png 2x" data-file-width="205" data-file-height="206" /></a></span><br /><i>t</i> </td></tr> <tr> <td>1 </td> <td>6 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7605c575fe5a9100156fabebfadca202a26d0612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&1\\0&6&0\\0&4&0\end{bmatrix}}}" /></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_mC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Conway_mC.png/100px-Conway_mC.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Conway_mC.png/150px-Conway_mC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0b/Conway_mC.png 2x" data-file-width="193" data-file-height="189" /></a></span><br /> <i>m</i> = <i>m</i><sub>1</sub> = <i>kj</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_bC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Conway_bC.png/100px-Conway_bC.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Conway_bC.png/150px-Conway_bC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7a/Conway_bC.png 2x" data-file-width="188" data-file-height="185" /></a></span><br /> <i>b</i> = <i>b</i><sub>1</sub> = <i>ta</i> </td></tr> <tr> <td>2 </td> <td>9 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ba05f773c9169dbb60f3392cca1cd45f26218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_m3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Conway_m3C.png/100px-Conway_m3C.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Conway_m3C.png/150px-Conway_m3C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Conway_m3C.png/200px-Conway_m3C.png 2x" data-file-width="389" data-file-height="389" /></a></span><br /><i>m</i><sub>2</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_m3dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Conway_m3dC.png/100px-Conway_m3dC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Conway_m3dC.png/150px-Conway_m3dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Conway_m3dC.png/200px-Conway_m3dC.png 2x" data-file-width="386" data-file-height="389" /></a></span><br /><i>m</i><sub>2</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_b3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Conway_b3C.png/100px-Conway_b3C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Conway_b3C.png/150px-Conway_b3C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Conway_b3C.png/200px-Conway_b3C.png 2x" data-file-width="381" data-file-height="390" /></a></span><br /><i>b</i><sub>2</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dm3dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Conway_dm3dC.png/100px-Conway_dm3dC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Conway_dm3dC.png/150px-Conway_dm3dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Conway_dm3dC.png/200px-Conway_dm3dC.png 2x" data-file-width="371" data-file-height="384" /></a></span><br /><i>b</i><sub>2</sub><i>d</i> </td></tr> <tr> <td>3 </td> <td>12 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&3&1\\0&12&0\\0&8&0\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> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&1\\0&12&0\\0&8&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e11542c6d00c06caee0805d2a8381b47d9b9c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&3&1\\0&12&0\\0&8&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_m4C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Conway_m4C.png/120px-Conway_m4C.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Conway_m4C.png/250px-Conway_m4C.png 1.5x" data-file-width="401" data-file-height="401" /></a></span><br /><i>m</i><sub>3</sub> </td> <td><i>m</i><sub>3</sub><i>d</i> </td> <td><i>b</i><sub>3</sub> </td> <td><i>b</i><sub>3</sub><i>d</i> </td></tr> <tr> <td><i>n</i> </td> <td>3<i>n</i>+3 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&n&1\\0&3n+3&0\\0&2n+2&0\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> <mn>1</mn> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&n&1\\0&3n+3&0\\0&2n+2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044f8d26c2534e02c3a4ba718131b04494df7cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.382ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&n&1\\0&3n+3&0\\0&2n+2&0\end{bmatrix}}}" /></span> </td> <td><i>m<sub>n</sub></i> </td> <td><i>m<sub>n</sub>d</i> </td> <td><i>b<sub>n</sub></i> </td> <td><i>b<sub>n</sub>d</i> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Medial">Medial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=10" title="Edit section: Medial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Medial is like meta, except it does not add edges from the center to each seed vertex. The index 1 form is identical to Conway's ortho and expand operators: expand is also called <a href="/wiki/Cantellation_(geometry)" title="Cantellation (geometry)">cantellation</a> and <a href="/wiki/Expansion_(geometry)" title="Expansion (geometry)">expansion</a>. Note that <i>o</i> and <i>e</i> have their own indexed forms, described below. Also note that some implementations start indexing at 0 instead of 1.<sup id="cite_ref-Antiprism_4-4" class="reference"><a href="#cite_note-Antiprism-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable sortable" style="text-align: center"> <caption>Medial operators </caption> <tbody><tr> <th>n</th> <th>Edge<br />factor</th> <th>Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span></th> <th>x</th> <th>xd</th> <th>dx</th> <th>dxd </th></tr> <tr> <td>1 </td> <td>4 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0d385c1d46c4a5b8a4fb36c0abdec5f26a3800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_oC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Conway_oC.png/120px-Conway_oC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Conway_oC.png/150px-Conway_oC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/38/Conway_oC.png 2x" data-file-width="189" data-file-height="187" /></a></span><br /> <i>M</i><sub>1</sub> = <i>o</i> = <i>jj</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_eC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_eC.png/100px-Conway_eC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_eC.png/150px-Conway_eC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/78/Conway_eC.png 2x" data-file-width="187" data-file-height="194" /></a></span><br /> <i>e</i> = <i>aa</i> </td></tr> <tr> <td>2 </td> <td>7 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c1c28733dc91c447129dbd0798788f2dcbb4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1\\0&7&0\\0&4&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_MC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Conway_MC.png/100px-Conway_MC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Conway_MC.png/150px-Conway_MC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Conway_MC.png/200px-Conway_MC.png 2x" data-file-width="389" data-file-height="389" /></a></span><br /><b>Medial</b>: <i>M</i> = <i>M</i><sub>2</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_MdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Conway_MdC.png/120px-Conway_MdC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Conway_MdC.png/250px-Conway_MdC.png 1.5x" data-file-width="381" data-file-height="392" /></a></span><br /><i>Md</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dMC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Conway_dMC.png/100px-Conway_dMC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Conway_dMC.png/150px-Conway_dMC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Conway_dMC.png/200px-Conway_dMC.png 2x" data-file-width="386" data-file-height="391" /></a></span><br /><i>dM</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dMdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Conway_dMdC.png/100px-Conway_dMdC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Conway_dMdC.png/150px-Conway_dMdC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Conway_dMdC.png/200px-Conway_dMdC.png 2x" data-file-width="383" data-file-height="380" /></a></span><br /><i>dMd</i> </td></tr> <tr> <td><i>n</i> </td> <td>3<i>n</i>+1 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&n&1\\0&3n+1&0\\0&2n&0\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> <mn>1</mn> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>n</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&n&1\\0&3n+1&0\\0&2n&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98a4cd613294ea4da26dfe9e7f02881d102e336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.382ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&n&1\\0&3n+1&0\\0&2n&0\end{bmatrix}}}" /></span> </td> <td><i>M<sub>n</sub></i> </td> <td><i>M<sub>n</sub>d</i> </td> <td><i>dM<sub>n</sub></i> </td> <td><i>dM<sub>n</sub>d</i> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Goldberg-Coxeter">Goldberg-Coxeter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=11" title="Edit section: Goldberg-Coxeter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Goldberg-Coxeter (GC) Conway operators are two infinite families of operators that are an extension of the <a href="/wiki/Goldberg-Coxeter_construction" class="mw-redirect" title="Goldberg-Coxeter construction">Goldberg-Coxeter construction</a>.<sup id="cite_ref-Deza2004_16-0" class="reference"><a href="#cite_note-Deza2004-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Deza2015_17-0" class="reference"><a href="#cite_note-Deza2015-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The GC construction can be thought of as taking a triangular section of a triangular lattice, or a square section of a square lattice, and laying that over each face of the polyhedron. This construction can be extended to any face by identifying the chambers of the triangle or square (the "master polygon").<sup id="cite_ref-Brinkmann_7-2" class="reference"><a href="#cite_note-Brinkmann-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Operators in the triangular family can be used to produce the <a href="/wiki/Goldberg_polyhedra" class="mw-redirect" title="Goldberg polyhedra">Goldberg polyhedra</a> and <a href="/wiki/Geodesic_polyhedra" class="mw-redirect" title="Geodesic polyhedra">geodesic polyhedra</a>: see <a href="/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra" title="List of geodesic polyhedra and Goldberg polyhedra">List of geodesic polyhedra and Goldberg polyhedra</a> for formulas. </p><p>The two families are the triangular GC family, <i>c<sub>a,b</sub></i> and <i>u<sub>a,b</sub></i>, and the quadrilateral GC family, <i>e<sub>a,b</sub></i> and <i>o<sub>a,b</sub></i>. Both the GC families are indexed by two integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06f20f72b40654833aff35ad637c3bb7c36fe5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.491ex; height:2.343ex;" alt="{\displaystyle a\geq 1}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb90a1049b38d6a352b9bde75bda4cd2f76515dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.258ex; height:2.343ex;" alt="{\displaystyle b\geq 0}" /></span>. They possess many nice qualities: </p> <ul><li>The indexes of the families have a relationship with certain <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domains</a> over the complex numbers: the <a href="/wiki/Eisenstein_integers" class="mw-redirect" title="Eisenstein integers">Eisenstein integers</a> for the triangular GC family, and the <a href="/wiki/Gaussian_integers" class="mw-redirect" title="Gaussian integers">Gaussian integers</a> for the quadrilateral GC family.</li> <li>Operators in the <i>x</i> and <i>dxd</i> columns within the same family commute with each other.</li></ul> <p>The operators are divided into three classes (examples are written in terms of <i>c</i> but apply to all 4 operators): </p> <ul><li>Class I: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19206e7d4dab695ccb34c502eff0741e98dbdfc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}" /></span>⁠</span>. Achiral, preserves original edges. Can be written with the zero index suppressed, e.g. <i>c</i><sub><i>a</i>,0</sub> = <i>c<sub>a</sub></i>.</li> <li>Class II: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" /></span>⁠</span>. Also achiral. Can be decomposed as <i>c<sub>a,a</sub></i> = <i>c<sub>a</sub>c</i><sub>1,1</sub></li> <li>Class III: All other operators. These are chiral, and <i>c<sub>a,b</sub></i> and <i>c<sub>b,a</sub></i> are the chiral pairs of each other.</li></ul> <p>Of the original Conway operations, the only ones that do not fall into the GC family are <i>g</i> and <i>s</i> (gyro and snub). Meta and bevel (<i>m</i> and <i>b</i>) can be expressed in terms of one operator from the triangular family and one from the quadrilateral family. </p> <div class="mw-heading mw-heading4"><h4 id="Triangular">Triangular</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=12" title="Edit section: Triangular"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" style="text-align: center"> <caption>Triangular Goldberg-Coxeter operators </caption> <tbody><tr> <th>a</th> <th>b</th> <th>Class</th> <th>Edge factor <br />T = a<sup>2</sup> + ab + b<sup>2</sup></th> <th>Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span></th> <th>Master triangle</th> <th>x</th> <th>xd</th> <th>dx</th> <th>dxd </th></tr> <tr> <td>1 </td> <td>0 </td> <td>I </td> <td>1 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917dc504a6780a695d578a7b216036af7e49c506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_01_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Subdivided_triangle_01_00.svg/120px-Subdivided_triangle_01_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Subdivided_triangle_01_00.svg/150px-Subdivided_triangle_01_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Subdivided_triangle_01_00.svg/200px-Subdivided_triangle_01_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/100px-Conway_C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/150px-Conway_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/200px-Conway_C.png 2x" data-file-width="371" data-file-height="377" /></a></span><br /><i>u</i><sub>1</sub> = <i>S</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Conway_dC.png/100px-Conway_dC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Conway_dC.png/150px-Conway_dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6d/Conway_dC.png 2x" data-file-width="190" data-file-height="192" /></a></span><br /><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/100px-Conway_C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/150px-Conway_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/200px-Conway_C.png 2x" data-file-width="371" data-file-height="377" /></a></span><br /><i>c</i><sub>1</sub> = <i>S</i> </td></tr> <tr> <td>2 </td> <td>0 </td> <td>I </td> <td>4 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&0\\0&4&0\\0&2&1\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&0\\0&4&0\\0&2&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb390a1a408121f11943f8b0ffedf8aa8065aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&0\\0&4&0\\0&2&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_02_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Subdivided_triangle_02_00.svg/100px-Subdivided_triangle_02_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Subdivided_triangle_02_00.svg/150px-Subdivided_triangle_02_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Subdivided_triangle_02_00.svg/200px-Subdivided_triangle_02_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_uC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_uC.png/120px-Conway_uC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Conway_uC.png/250px-Conway_uC.png 1.5x" data-file-width="389" data-file-height="398" /></a></span><br /><i>u</i><sub>2</sub> = <i>u</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dcC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Conway_dcC.png/100px-Conway_dcC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Conway_dcC.png/150px-Conway_dcC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/70/Conway_dcC.png 2x" data-file-width="195" data-file-height="198" /></a></span><br /><i>dc</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_duC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_duC.png/100px-Conway_duC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_duC.png/150px-Conway_duC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_duC.png/200px-Conway_duC.png 2x" data-file-width="382" data-file-height="389" /></a></span><br /><i>du</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_cC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/100px-Conway_cC.png" decoding="async" width="100" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Conway_cC.png/150px-Conway_cC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/47/Conway_cC.png 2x" data-file-width="188" data-file-height="183" /></a></span><br /><i>c</i><sub>2</sub> = <i>c</i> </td></tr> <tr> <td>3 </td> <td>0 </td> <td>I </td> <td>9 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ba05f773c9169dbb60f3392cca1cd45f26218c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1\\0&9&0\\0&6&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_03_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Subdivided_triangle_03_00.svg/100px-Subdivided_triangle_03_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Subdivided_triangle_03_00.svg/150px-Subdivided_triangle_03_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Subdivided_triangle_03_00.svg/200px-Subdivided_triangle_03_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_ktC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_ktC.png/100px-Conway_ktC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_ktC.png/150px-Conway_ktC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_ktC.png/200px-Conway_ktC.png 2x" data-file-width="405" data-file-height="406" /></a></span><br /><i>u</i><sub>3</sub> = <i>nn</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dtkC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_dtkC.png/100px-Conway_dtkC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_dtkC.png/150px-Conway_dtkC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Conway_dtkC.png/200px-Conway_dtkC.png 2x" data-file-width="487" data-file-height="495" /></a></span><br /><i>nk</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dktC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Conway_dktC.png/100px-Conway_dktC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Conway_dktC.png/150px-Conway_dktC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Conway_dktC.png/200px-Conway_dktC.png 2x" data-file-width="374" data-file-height="378" /></a></span><br /><i>zt</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_tkC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Conway_tkC.png/120px-Conway_tkC.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Conway_tkC.png/250px-Conway_tkC.png 1.5x" data-file-width="404" data-file-height="389" /></a></span><br /><i>c</i><sub>3</sub> = <i>zz</i> </td></tr> <tr> <td>4 </td> <td>0 </td> <td>I </td> <td>16 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&5&0\\0&16&0\\0&10&1\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> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&5&0\\0&16&0\\0&10&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38ab86a9e899fde63f384c3f0229ed64f47f76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&5&0\\0&16&0\\0&10&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_04_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Subdivided_triangle_04_00.svg/100px-Subdivided_triangle_04_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Subdivided_triangle_04_00.svg/150px-Subdivided_triangle_04_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Subdivided_triangle_04_00.svg/200px-Subdivided_triangle_04_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_u4C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Conway_u4C.png/100px-Conway_u4C.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Conway_u4C.png/150px-Conway_u4C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Conway_u4C.png/200px-Conway_u4C.png 2x" data-file-width="1085" data-file-height="1095" /></a></span><br /><i>u</i><sub>4</sub> = <i>uu</i> </td> <td><i>uud</i> = <i>dcc</i> </td> <td><i>duu</i> = <i>ccd</i> </td> <td><i>c</i><sub>4</sub> = <i>cc</i> </td></tr> <tr> <td>5 </td> <td>0 </td> <td>I </td> <td>25 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&8&0\\0&25&0\\0&16&1\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> <mn>1</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>25</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&8&0\\0&25&0\\0&16&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8280b6f0f9830d848d6ecee8ca2f2734011e420c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&8&0\\0&25&0\\0&16&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_05_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Subdivided_triangle_05_00.svg/100px-Subdivided_triangle_05_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Subdivided_triangle_05_00.svg/150px-Subdivided_triangle_05_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Subdivided_triangle_05_00.svg/200px-Subdivided_triangle_05_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_u5C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Conway_u5C.png/100px-Conway_u5C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Conway_u5C.png/150px-Conway_u5C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Conway_u5C.png/200px-Conway_u5C.png 2x" data-file-width="1070" data-file-height="1095" /></a></span><br /><i>u</i><sub>5</sub> </td> <td><i>u</i><sub>5</sub><i>d</i> = <i>dc</i><sub>5</sub> </td> <td><i>du</i><sub>5</sub> = <i>c</i><sub>5</sub><i>d</i> </td> <td><i>c</i><sub>5</sub> </td></tr> <tr> <td>6 </td> <td>0 </td> <td>I </td> <td>36 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&11&1\\0&36&0\\0&24&0\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> <mn>1</mn> </mtd> <mtd> <mn>11</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>36</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>24</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&11&1\\0&36&0\\0&24&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8174f784599808e5de63e5f6d6b8c734c2945e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&11&1\\0&36&0\\0&24&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_06_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Subdivided_triangle_06_00.svg/100px-Subdivided_triangle_06_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Subdivided_triangle_06_00.svg/150px-Subdivided_triangle_06_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Subdivided_triangle_06_00.svg/200px-Subdivided_triangle_06_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_u6C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Conway_u6C.png/120px-Conway_u6C.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Conway_u6C.png/250px-Conway_u6C.png 1.5x" data-file-width="1052" data-file-height="1091" /></a></span><br /><i>u</i><sub>6</sub> = <i>unn</i> </td> <td><i>unk</i> </td> <td><i>czt</i> </td> <td><i>u</i><sub>6</sub> = <i>czz</i> </td></tr> <tr> <td>7 </td> <td>0 </td> <td>I </td> <td>49 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&16&0\\0&49&0\\0&32&1\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> <mn>1</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>49</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&16&0\\0&49&0\\0&32&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c453504f60d003e85ad42cdd334da7165217a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&16&0\\0&49&0\\0&32&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_07_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Subdivided_triangle_07_00.svg/120px-Subdivided_triangle_07_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Subdivided_triangle_07_00.svg/150px-Subdivided_triangle_07_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Subdivided_triangle_07_00.svg/200px-Subdivided_triangle_07_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_u7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Conway_u7.png/100px-Conway_u7.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Conway_u7.png/150px-Conway_u7.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Conway_u7.png/200px-Conway_u7.png 2x" data-file-width="1082" data-file-height="1067" /></a></span><br /><i>u</i><sub>7</sub> = <i>u</i><sub>2,1</sub><i>u</i><sub>1,2</sub> = <i>vrv</i> </td> <td><i>vrvd</i> = <i>dwrw</i> </td> <td><i>dvrv</i> = <i>wrwd</i> </td> <td><i>c</i><sub>7</sub> = <i>c</i><sub>2,1</sub><i>c</i><sub>1,2</sub> = <i>wrw</i> </td></tr> <tr> <td>8 </td> <td>0 </td> <td>I </td> <td>64 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&21&0\\0&64&0\\0&42&1\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> <mn>1</mn> </mtd> <mtd> <mn>21</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>64</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>42</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&21&0\\0&64&0\\0&42&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c270dba22f41d980b72fd0d0ae5914cfd7738c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&21&0\\0&64&0\\0&42&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_08_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Subdivided_triangle_08_00.svg/120px-Subdivided_triangle_08_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Subdivided_triangle_08_00.svg/150px-Subdivided_triangle_08_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Subdivided_triangle_08_00.svg/200px-Subdivided_triangle_08_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_u8C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Conway_u8C.png/120px-Conway_u8C.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Conway_u8C.png/250px-Conway_u8C.png 1.5x" data-file-width="1050" data-file-height="1086" /></a></span><br /><i>u</i><sub>8</sub> = <i>u</i><sup>3</sup> </td> <td><i>u</i><sup>3</sup><i>d</i> = <i>dc</i><sup>3</sup> </td> <td><i>du</i><sup>3</sup> = <i>c</i><sup>3</sup><i>d</i> </td> <td><i>c</i><sub>8</sub> = <i>c</i><sup>3</sup> </td></tr> <tr> <td>9 </td> <td>0 </td> <td>I </td> <td>81 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&26&1\\0&81&0\\0&54&0\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> <mn>1</mn> </mtd> <mtd> <mn>26</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>81</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>54</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&26&1\\0&81&0\\0&54&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b8a1f80b35ec1b9689e5204ab322f76a21eaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&26&1\\0&81&0\\0&54&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_09_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Subdivided_triangle_09_00.svg/100px-Subdivided_triangle_09_00.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Subdivided_triangle_09_00.svg/150px-Subdivided_triangle_09_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Subdivided_triangle_09_00.svg/200px-Subdivided_triangle_09_00.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_u9C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Conway_u9C.png/100px-Conway_u9C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Conway_u9C.png/150px-Conway_u9C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Conway_u9C.png/200px-Conway_u9C.png 2x" data-file-width="1062" data-file-height="1088" /></a></span><br /><i>u</i><sub>9</sub> = <i>n</i><sup>4</sup> </td> <td><i>n</i><sup>3</sup><i>k</i> = <i>kz</i><sup>3</sup> </td> <td><i>tn</i><sup>3</sup> = <i>z</i><sup>3</sup><i>t</i> </td> <td><i>c</i><sub>9</sub> = <i>z</i><sup>4</sup> </td></tr> <tr> <td>1 </td> <td>1 </td> <td>II </td> <td>3 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9419a43c4ffce9c1e51a5e52bee7c9a445ca72d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&1\\0&3&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_01_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Subdivided_triangle_01_01.svg/100px-Subdivided_triangle_01_01.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Subdivided_triangle_01_01.svg/150px-Subdivided_triangle_01_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Subdivided_triangle_01_01.svg/200px-Subdivided_triangle_01_01.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kdC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Conway_kdC.png/120px-Conway_kdC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Conway_kdC.png/250px-Conway_kdC.png 1.5x" data-file-width="372" data-file-height="377" /></a></span><br /><i>u</i><sub>1,1</sub> = <i>n</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_kC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/120px-Conway_kC.png" decoding="async" width="100" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Conway_kC.png/150px-Conway_kC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b7/Conway_kC.png 2x" data-file-width="191" data-file-height="205" /></a></span><br /><i>k</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_tC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/100px-Conway_tC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/150px-Conway_tC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Conway_tC.png/200px-Conway_tC.png 2x" data-file-width="205" data-file-height="206" /></a></span><br /><i>t</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dkC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/100px-Conway_dkC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/150px-Conway_dkC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Conway_dkC.png/200px-Conway_dkC.png 2x" data-file-width="415" data-file-height="419" /></a></span><br /><i>c</i><sub>1,1</sub> = <i>z</i> </td></tr> <tr> <td>2 </td> <td>1 </td> <td>III </td> <td>7 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0b3ccc2dbba110409c7229b97e402ec7889fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&7&0\\0&4&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_02_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Subdivided_triangle_02_01.svg/100px-Subdivided_triangle_02_01.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Subdivided_triangle_02_01.svg/150px-Subdivided_triangle_02_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Subdivided_triangle_02_01.svg/200px-Subdivided_triangle_02_01.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>v</i> = <i>u</i><sub>2,1</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_dwC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Conway_dwC.png/100px-Conway_dwC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Conway_dwC.png/150px-Conway_dwC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Conway_dwC.png/200px-Conway_dwC.png 2x" data-file-width="388" data-file-height="392" /></a></span><br /><i>vd</i> = <i>dw</i> </td> <td><i>dv</i> = <i>wd</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_wC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Conway_wC.png/100px-Conway_wC.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Conway_wC.png/150px-Conway_wC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Conway_wC.png/200px-Conway_wC.png 2x" data-file-width="397" data-file-height="398" /></a></span><br /><i>w</i> = <i>c</i><sub>2,1</sub> </td></tr> <tr> <td>3 </td> <td>1 </td> <td>III </td> <td>13 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&4&0\\0&13&0\\0&8&1\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> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>13</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&4&0\\0&13&0\\0&8&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7713f3291cb2909a44116cb8a5581fbe253fb7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&4&0\\0&13&0\\0&8&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_03_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Subdivided_triangle_03_01.svg/120px-Subdivided_triangle_03_01.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Subdivided_triangle_03_01.svg/150px-Subdivided_triangle_03_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Subdivided_triangle_03_01.svg/200px-Subdivided_triangle_03_01.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>3,1</sub> </td> <td><i>u</i><sub>3,1</sub><i>d</i> = <i>dc</i><sub>3,1</sub> </td> <td><i>du</i><sub>3,1</sub> = <i>c</i><sub>3,1</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Conway_w3C.png/100px-Conway_w3C.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Conway_w3C.png/150px-Conway_w3C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Conway_w3C.png/200px-Conway_w3C.png 2x" data-file-width="1104" data-file-height="1089" /></a></span><br /><i>c</i><sub>3,1</sub> </td></tr> <tr> <td>3 </td> <td>2 </td> <td>III </td> <td>19 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&6&0\\0&19&0\\0&12&1\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> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>19</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&6&0\\0&19&0\\0&12&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8f2ea0e945c4758726280d08eb73eea9e93611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&6&0\\0&19&0\\0&12&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_03_02.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Subdivided_triangle_03_02.svg/100px-Subdivided_triangle_03_02.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Subdivided_triangle_03_02.svg/150px-Subdivided_triangle_03_02.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Subdivided_triangle_03_02.svg/200px-Subdivided_triangle_03_02.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>3,2</sub> </td> <td><i>u</i><sub>3,2</sub><i>d</i> = <i>dc</i><sub>3,2</sub> </td> <td><i>du</i><sub>3,2</sub> = <i>c</i><sub>3,2</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w3-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Conway_w3-2.png/120px-Conway_w3-2.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Conway_w3-2.png/250px-Conway_w3-2.png 1.5x" data-file-width="392" data-file-height="395" /></a></span><br /><i>c</i><sub>3,2</sub> </td></tr> <tr> <td>4 </td> <td>3 </td> <td>III </td> <td>37 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&12&0\\0&37&0\\0&24&1\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> <mn>1</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>37</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>24</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&12&0\\0&37&0\\0&24&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb84afe60b558f165d3dc4168d5b9bc72ce328a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&12&0\\0&37&0\\0&24&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_04_03.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Subdivided_triangle_04_03.svg/120px-Subdivided_triangle_04_03.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Subdivided_triangle_04_03.svg/150px-Subdivided_triangle_04_03.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Subdivided_triangle_04_03.svg/200px-Subdivided_triangle_04_03.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>4,3</sub> </td> <td><i>u</i><sub>4,3</sub><i>d</i> = <i>dc</i><sub>4,3</sub> </td> <td><i>du</i><sub>4,3</sub> = <i>c</i><sub>4,3</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w4-3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Conway_w4-3C.png/120px-Conway_w4-3C.png" decoding="async" width="100" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Conway_w4-3C.png/250px-Conway_w4-3C.png 1.5x" data-file-width="1103" data-file-height="1065" /></a></span><br /><i>c</i><sub>4,3</sub> </td></tr> <tr> <td>5 </td> <td>4 </td> <td>III </td> <td>61 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&20&0\\0&61&0\\0&40&1\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> <mn>1</mn> </mtd> <mtd> <mn>20</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>61</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>40</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&20&0\\0&61&0\\0&40&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2879e4769e2468251c296c33da0d55091e9d494c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&20&0\\0&61&0\\0&40&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_05_04.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Subdivided_triangle_05_04.svg/100px-Subdivided_triangle_05_04.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Subdivided_triangle_05_04.svg/150px-Subdivided_triangle_05_04.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Subdivided_triangle_05_04.svg/200px-Subdivided_triangle_05_04.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>5,4</sub> </td> <td><i>u</i><sub>5,4</sub><i>d</i> = <i>dc</i><sub>5,4</sub> </td> <td><i>du</i><sub>5,4</sub> = <i>c</i><sub>5,4</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w5-4C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Conway_w5-4C.png/100px-Conway_w5-4C.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Conway_w5-4C.png/150px-Conway_w5-4C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Conway_w5-4C.png/200px-Conway_w5-4C.png 2x" data-file-width="1098" data-file-height="1104" /></a></span><br /><i>c</i><sub>5,4</sub> </td></tr> <tr> <td>6 </td> <td>5 </td> <td>III </td> <td>91 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&30&0\\0&91&0\\0&60&1\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> <mn>1</mn> </mtd> <mtd> <mn>30</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>91</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>60</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&30&0\\0&91&0\\0&60&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d44e64e413584f5365306e68b2a204745e1874e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&30&0\\0&91&0\\0&60&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_06_05.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Subdivided_triangle_06_05.svg/120px-Subdivided_triangle_06_05.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Subdivided_triangle_06_05.svg/150px-Subdivided_triangle_06_05.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Subdivided_triangle_06_05.svg/200px-Subdivided_triangle_06_05.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>6,5</sub> = <i>u</i><sub>1,2</sub><i>u</i><sub>1,3</sub> </td> <td><i>u</i><sub>6,5</sub><i>d</i> = <i>dc</i><sub>6,5</sub> </td> <td><i>du</i><sub>6,5</sub> = <i>c</i><sub>6,5</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w6-5C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Conway_w6-5C.png/100px-Conway_w6-5C.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Conway_w6-5C.png/150px-Conway_w6-5C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Conway_w6-5C.png/200px-Conway_w6-5C.png 2x" data-file-width="1073" data-file-height="1089" /></a></span><br /><i>c</i><sub>6,5</sub>=<i>c</i><sub>1,2</sub><i>c</i><sub>1,3</sub> </td></tr> <tr> <td>7 </td> <td>6 </td> <td>III </td> <td>127 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&42&0\\0&127&0\\0&84&1\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> <mn>1</mn> </mtd> <mtd> <mn>42</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>127</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>84</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&42&0\\0&127&0\\0&84&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2157c0f4aeb50f0d5be34ffd97d7128a960923cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.31ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&42&0\\0&127&0\\0&84&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_07_06.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Subdivided_triangle_07_06.svg/100px-Subdivided_triangle_07_06.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Subdivided_triangle_07_06.svg/150px-Subdivided_triangle_07_06.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Subdivided_triangle_07_06.svg/200px-Subdivided_triangle_07_06.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>7,6</sub> </td> <td><i>u</i><sub>7,6</sub><i>d</i> = <i>dc</i><sub>7,6</sub> </td> <td><i>du</i><sub>7,6</sub> = <i>c</i><sub>7,6</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w7C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Conway_w7C.png/120px-Conway_w7C.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Conway_w7C.png/250px-Conway_w7C.png 1.5x" data-file-width="1200" data-file-height="1200" /></a></span><br /><i>c</i><sub>7,6</sub> </td></tr> <tr> <td>8 </td> <td>7 </td> <td>III </td> <td>169 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&56&0\\0&169&0\\0&112&1\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> <mn>1</mn> </mtd> <mtd> <mn>56</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>169</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>112</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&56&0\\0&169&0\\0&112&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095b30746d684911a5c206efae093d099f2993f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.31ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&56&0\\0&169&0\\0&112&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_08_07.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Subdivided_triangle_08_07.svg/100px-Subdivided_triangle_08_07.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Subdivided_triangle_08_07.svg/150px-Subdivided_triangle_08_07.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Subdivided_triangle_08_07.svg/200px-Subdivided_triangle_08_07.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>8,7</sub> = <i>u</i><sub>3,1</sub><sup>2</sup> </td> <td><i>u</i><sub>8,7</sub><i>d</i> = <i>dc</i><sub>8,7</sub> </td> <td><i>du</i><sub>8,7</sub> = <i>c</i><sub>8,7</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w8C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Conway_w8C.png/100px-Conway_w8C.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Conway_w8C.png/150px-Conway_w8C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Conway_w8C.png/200px-Conway_w8C.png 2x" data-file-width="1200" data-file-height="1200" /></a></span><br /><i>c</i><sub>8,7</sub> = <i>c</i><sub>3,1</sub><sup>2</sup> </td></tr> <tr> <td>9 </td> <td>8 </td> <td>III </td> <td>217 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&72&0\\0&217&0\\0&144&1\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> <mn>1</mn> </mtd> <mtd> <mn>72</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>217</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>144</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&72&0\\0&217&0\\0&144&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/553de36eee1353b5b1f2182fde8c767a074bd1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.31ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&72&0\\0&217&0\\0&144&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_triangle_09_08.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Subdivided_triangle_09_08.svg/100px-Subdivided_triangle_09_08.svg.png" decoding="async" width="100" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Subdivided_triangle_09_08.svg/150px-Subdivided_triangle_09_08.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Subdivided_triangle_09_08.svg/200px-Subdivided_triangle_09_08.svg.png 2x" data-file-width="182" data-file-height="158" /></a></span> </td> <td><i>u</i><sub>9,8</sub> = <i>u</i><sub>2,1</sub><i>u</i><sub>5,1</sub> </td> <td><i>u</i><sub>9,8</sub><i>d</i> = <i>dc</i><sub>9,8</sub> </td> <td><i>du</i><sub>9,8</sub> = <i>c</i><sub>9,8</sub><i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_w9C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Conway_w9C.png/100px-Conway_w9C.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Conway_w9C.png/150px-Conway_w9C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Conway_w9C.png/200px-Conway_w9C.png 2x" data-file-width="1200" data-file-height="1200" /></a></span><br /><i>c</i><sub>9,8</sub> = <i>c</i><sub>2,1</sub><i>c</i><sub>5,1</sub> </td></tr> <tr> <td colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\equiv b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859529d640f85a2c3bf2847f61a842ba3a5753ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a\equiv b}" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\mathrm {mod} \ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\mathrm {mod} \ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3424c0a3da3b0ec3892141e63befd406a907af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.524ex; height:2.843ex;" alt="{\displaystyle \ (\mathrm {mod} \ 3)}" /></span> </td> <td>I, II, or III </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\equiv 0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≡</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\equiv 0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599e254fb5a49fee5a8929b9863360f80e9175f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.176ex;" alt="{\displaystyle T\equiv 0\ }" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathrm {mod} \ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {mod} \ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b2f5e857a8d733f998cea9ef6362e947391789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.943ex; height:2.843ex;" alt="{\displaystyle (\mathrm {mod} \ 3)}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&{\frac {T}{3}}-1&1\\0&T&0\\0&{\frac {2}{3}}T&0\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> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>T</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>T</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&{\frac {T}{3}}-1&1\\0&T&0\\0&{\frac {2}{3}}T&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24478d70127e0f99e61ac71daabd0c4897fb5e84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:16.818ex; height:10.843ex;" alt="{\displaystyle {\begin{bmatrix}1&{\frac {T}{3}}-1&1\\0&T&0\\0&{\frac {2}{3}}T&0\end{bmatrix}}}" /></span> </td> <td>... </td> <td><i>u<sub>a,b</sub></i> </td> <td><i>u<sub>a,b</sub>d</i> = <i>dc<sub>a,b</sub></i> </td> <td><i>du<sub>a,b</sub></i> = <i>c<sub>a,b</sub>d</i> </td> <td><i>c<sub>a,b</sub></i> </td></tr> <tr> <td colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\not \equiv b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≢</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\not \equiv b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08277b01c0fff9691e6bc25c2ca7663d9059ad07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.327ex; height:2.676ex;" alt="{\displaystyle a\not \equiv b}" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\mathrm {mod} \ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\mathrm {mod} \ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3424c0a3da3b0ec3892141e63befd406a907af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.524ex; height:2.843ex;" alt="{\displaystyle \ (\mathrm {mod} \ 3)}" /></span> </td> <td>I or III </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\equiv 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≡</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\equiv 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a88b8226848397a145be692c66c81b1288c3f2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T\equiv 1}" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\mathrm {mod} \ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\mathrm {mod} \ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3424c0a3da3b0ec3892141e63befd406a907af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.524ex; height:2.843ex;" alt="{\displaystyle \ (\mathrm {mod} \ 3)}" /></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&{\frac {T-1}{3}}&0\\0&T&0\\0&2{\frac {T-1}{3}}&1\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> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>T</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&{\frac {T-1}{3}}&0\\0&T&0\\0&2{\frac {T-1}{3}}&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59dd9f533ef5f14e4433a5a96ce3a4f36d4dc2bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:16.078ex; height:11.176ex;" alt="{\displaystyle {\begin{bmatrix}1&{\frac {T-1}{3}}&0\\0&T&0\\0&2{\frac {T-1}{3}}&1\end{bmatrix}}}" /></span> </td> <td>... </td> <td><i>u<sub>a,b</sub></i> </td> <td><i>u<sub>a,b</sub>d</i> = <i>dc<sub>a,b</sub></i> </td> <td><i>du<sub>a,b</sub></i> = <i>c<sub>a,b</sub>d</i> </td> <td><i>c<sub>a,b</sub></i> </td></tr></tbody></table> <p>By basic number theory, for any values of <i>a</i> and <i>b</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\not \equiv 2\ (\mathrm {mod} \ 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≢</mo> <mn>2</mn> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\not \equiv 2\ (\mathrm {mod} \ 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f74fb0c2b48ab3d6fb27dafc0db4159f7e286b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.422ex; height:2.843ex;" alt="{\displaystyle T\not \equiv 2\ (\mathrm {mod} \ 3)}" /></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Quadrilateral">Quadrilateral</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=13" title="Edit section: Quadrilateral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" style="text-align: center"> <caption>Quadrilateral Goldberg-Coxeter operators </caption> <tbody><tr> <th>a</th> <th>b</th> <th>Class</th> <th>Edge factor <br />T = a<sup>2</sup> + b<sup>2</sup></th> <th>Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc44dadd8d20d808ce7ce877a7452b8fd171bec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {M} _{x}}" /></span></th> <th>Master square</th> <th>x</th> <th>xd</th> <th>dx</th> <th>dxd </th></tr> <tr> <td>1 </td> <td>0 </td> <td>I </td> <td>1 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917dc504a6780a695d578a7b216036af7e49c506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_01_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Subdivided_square_01_00.svg/100px-Subdivided_square_01_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Subdivided_square_01_00.svg/150px-Subdivided_square_01_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Subdivided_square_01_00.svg/200px-Subdivided_square_01_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/100px-Conway_C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/150px-Conway_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/200px-Conway_C.png 2x" data-file-width="371" data-file-height="377" /></a></span><br /><i>o</i><sub>1</sub> = <i>S</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_dC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Conway_dC.png/100px-Conway_dC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Conway_dC.png/150px-Conway_dC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6d/Conway_dC.png 2x" data-file-width="190" data-file-height="192" /></a></span><br /><i>e</i><sub>1</sub> = <i>d</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/100px-Conway_C.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/150px-Conway_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Conway_C.png/200px-Conway_C.png 2x" data-file-width="371" data-file-height="377" /></a></span><br /><i>o</i><sub>1</sub> = <i>dd</i> = <i>S</i> </td></tr> <tr> <td>2 </td> <td>0 </td> <td>I </td> <td>4 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\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> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0d385c1d46c4a5b8a4fb36c0abdec5f26a3800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&1\\0&4&0\\0&2&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_02_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Subdivided_square_02_00.svg/120px-Subdivided_square_02_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Subdivided_square_02_00.svg/150px-Subdivided_square_02_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Subdivided_square_02_00.svg/200px-Subdivided_square_02_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_oC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Conway_oC.png/120px-Conway_oC.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Conway_oC.png/150px-Conway_oC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/38/Conway_oC.png 2x" data-file-width="189" data-file-height="187" /></a></span><br /><i>o</i><sub>2</sub> = <i>o</i> = <i>j</i><sup>2</sup> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_eC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_eC.png/100px-Conway_eC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Conway_eC.png/150px-Conway_eC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/78/Conway_eC.png 2x" data-file-width="187" data-file-height="194" /></a></span><br /><i>e</i><sub>2</sub> = <i>e</i> = <i>a</i><sup>2</sup> </td></tr> <tr> <td>3 </td> <td>0 </td> <td>I </td> <td>9 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&4&0\\0&9&0\\0&4&1\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> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&4&0\\0&9&0\\0&4&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba508fc12eab19e0e69b9a790b5e67c5cffe23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&4&0\\0&9&0\\0&4&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_03_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Subdivided_square_03_00.svg/100px-Subdivided_square_03_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Subdivided_square_03_00.svg/150px-Subdivided_square_03_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Subdivided_square_03_00.svg/200px-Subdivided_square_03_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Conway_o3C.png/100px-Conway_o3C.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Conway_o3C.png/150px-Conway_o3C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Conway_o3C.png/200px-Conway_o3C.png 2x" data-file-width="386" data-file-height="384" /></a></span><br /><i>o</i><sub>3</sub> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_e3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Conway_e3C.png/100px-Conway_e3C.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Conway_e3C.png/150px-Conway_e3C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0f/Conway_e3C.png 2x" data-file-width="187" data-file-height="194" /></a></span><br /><i>e</i><sub>3</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o3C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Conway_o3C.png/100px-Conway_o3C.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Conway_o3C.png/150px-Conway_o3C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Conway_o3C.png/200px-Conway_o3C.png 2x" data-file-width="386" data-file-height="384" /></a></span><br /><i>o</i><sub>3</sub> </td></tr> <tr> <td>4 </td> <td>0 </td> <td>I </td> <td>16 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&7&1\\0&16&0\\0&8&0\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> <mn>1</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&7&1\\0&16&0\\0&8&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c9a9eb583bfb5c888af8527d5dc817de97c798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&7&1\\0&16&0\\0&8&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_04_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Subdivided_square_04_00.svg/100px-Subdivided_square_04_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Subdivided_square_04_00.svg/150px-Subdivided_square_04_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Subdivided_square_04_00.svg/200px-Subdivided_square_04_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_deeC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Conway_deeC.png/100px-Conway_deeC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Conway_deeC.png/150px-Conway_deeC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Conway_deeC.png/200px-Conway_deeC.png 2x" data-file-width="379" data-file-height="382" /></a></span><br /><i>o</i><sub>4</sub> = <i>oo</i> = <i>j</i><sup>4</sup> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_eeC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Conway_eeC.png/100px-Conway_eeC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Conway_eeC.png/150px-Conway_eeC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Conway_eeC.png/200px-Conway_eeC.png 2x" data-file-width="372" data-file-height="376" /></a></span><br /><i>e</i><sub>4</sub> = <i>ee</i> = <i>a</i><sup>4</sup> </td></tr> <tr> <td>5 </td> <td>0 </td> <td>I </td> <td>25 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&12&0\\0&25&0\\0&12&1\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> <mn>1</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>25</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&12&0\\0&25&0\\0&12&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccd1b2db71dde43a392fd4bd245af8c88627b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&12&0\\0&25&0\\0&12&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_05_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Subdivided_square_05_00.svg/120px-Subdivided_square_05_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Subdivided_square_05_00.svg/150px-Subdivided_square_05_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Subdivided_square_05_00.svg/200px-Subdivided_square_05_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o5C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_o5C.png/100px-Conway_o5C.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_o5C.png/150px-Conway_o5C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_o5C.png/200px-Conway_o5C.png 2x" data-file-width="392" data-file-height="395" /></a></span><br /><i>o</i><sub>5</sub> = <i>o</i><sub>2,1</sub><i>o</i><sub>1,2</sub> = <i>prp</i> </td> <td colspan="2"><i>e</i><sub>5</sub> = <i>e</i><sub>2,1</sub><i>e</i><sub>1,2</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o5C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_o5C.png/100px-Conway_o5C.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_o5C.png/150px-Conway_o5C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Conway_o5C.png/200px-Conway_o5C.png 2x" data-file-width="392" data-file-height="395" /></a></span><br /><i>o</i><sub>5</sub>= <i>dprpd</i> </td></tr> <tr> <td>6 </td> <td>0 </td> <td>I </td> <td>36 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&17&1\\0&36&0\\0&18&0\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> <mn>1</mn> </mtd> <mtd> <mn>17</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>36</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>18</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&17&1\\0&36&0\\0&18&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59611ae68a2bfc9b9d4955bb15606ac19bbb88b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&17&1\\0&36&0\\0&18&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_06_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Subdivided_square_06_00.svg/100px-Subdivided_square_06_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Subdivided_square_06_00.svg/150px-Subdivided_square_06_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Subdivided_square_06_00.svg/200px-Subdivided_square_06_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_o6C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Conway_o6C.png/100px-Conway_o6C.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Conway_o6C.png/150px-Conway_o6C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Conway_o6C.png/200px-Conway_o6C.png 2x" data-file-width="365" data-file-height="366" /></a></span><br /><i>o</i><sub>6</sub> = <i>o</i><sub>2</sub><i>o</i><sub>3</sub> </td> <td colspan="2"><i>e</i><sub>6</sub> = <i>e</i><sub>2</sub><i>e</i><sub>3</sub> </td></tr> <tr> <td>7 </td> <td>0 </td> <td>I </td> <td>49 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&24&0\\0&49&0\\0&24&1\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> <mn>1</mn> </mtd> <mtd> <mn>24</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>49</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>24</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&24&0\\0&49&0\\0&24&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be72edd90ac4d22fa6b6c45baa623b0b6a186c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&24&0\\0&49&0\\0&24&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_07_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Subdivided_square_07_00.svg/100px-Subdivided_square_07_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Subdivided_square_07_00.svg/150px-Subdivided_square_07_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Subdivided_square_07_00.svg/200px-Subdivided_square_07_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o7C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conway_o7C.png/100px-Conway_o7C.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conway_o7C.png/150px-Conway_o7C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conway_o7C.png/200px-Conway_o7C.png 2x" data-file-width="372" data-file-height="364" /></a></span><br /><i>o</i><sub>7</sub> </td> <td colspan="2"><i>e</i><sub>7</sub> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o7C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conway_o7C.png/100px-Conway_o7C.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conway_o7C.png/150px-Conway_o7C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Conway_o7C.png/200px-Conway_o7C.png 2x" data-file-width="372" data-file-height="364" /></a></span><br /><i>o</i><sub>7</sub> </td></tr> <tr> <td>8 </td> <td>0 </td> <td>I </td> <td>64 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&31&1\\0&64&0\\0&32&0\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> <mn>1</mn> </mtd> <mtd> <mn>31</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>64</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>32</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&31&1\\0&64&0\\0&32&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d45b2e4cd8221c9a43f9abc1f9176406447cbf78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&31&1\\0&64&0\\0&32&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_08_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Subdivided_square_08_00.svg/120px-Subdivided_square_08_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Subdivided_square_08_00.svg/150px-Subdivided_square_08_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Subdivided_square_08_00.svg/200px-Subdivided_square_08_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_o8C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Conway_o8C.png/100px-Conway_o8C.png" decoding="async" width="100" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Conway_o8C.png/150px-Conway_o8C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Conway_o8C.png/200px-Conway_o8C.png 2x" data-file-width="369" data-file-height="358" /></a></span><br /><i>o</i><sub>8</sub> = <i>o</i><sup>3</sup> = <i>j</i><sup>6</sup> </td> <td colspan="2"><i>e</i><sub>8</sub> = <i>e</i><sup>3</sup> = <i>a</i><sup>6</sup> </td></tr> <tr> <td>9 </td> <td>0 </td> <td>I </td> <td>81 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&40&0\\0&81&0\\0&40&1\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> <mn>1</mn> </mtd> <mtd> <mn>40</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>81</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>40</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&40&0\\0&81&0\\0&40&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1c8075b9c1d08151f2d6b97f320e59d50c0b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&40&0\\0&81&0\\0&40&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_09_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Subdivided_square_09_00.svg/100px-Subdivided_square_09_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Subdivided_square_09_00.svg/150px-Subdivided_square_09_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Subdivided_square_09_00.svg/200px-Subdivided_square_09_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o9C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Conway_o9C.png/100px-Conway_o9C.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Conway_o9C.png/150px-Conway_o9C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Conway_o9C.png/200px-Conway_o9C.png 2x" data-file-width="368" data-file-height="361" /></a></span><br /><i>o</i><sub>9</sub> = <i>o</i><sub>3</sub><sup>2</sup> </td> <td colspan="2"><br /><i>e</i><sub>9</sub> = <i>e</i><sub>3</sub><sup>2</sup> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_o9C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Conway_o9C.png/100px-Conway_o9C.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Conway_o9C.png/150px-Conway_o9C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Conway_o9C.png/200px-Conway_o9C.png 2x" data-file-width="368" data-file-height="361" /></a></span><br /><i>o</i><sub>9</sub> </td></tr> <tr> <td>10 </td> <td>0 </td> <td>I </td> <td>100 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&49&1\\0&100&0\\0&50&0\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> <mn>1</mn> </mtd> <mtd> <mn>49</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>100</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>50</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&49&1\\0&100&0\\0&50&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5172ff6bfd1a9c9f3be43f8e5ad78216aeb261f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.31ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&49&1\\0&100&0\\0&50&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_10_00.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Subdivided_square_10_00.svg/100px-Subdivided_square_10_00.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Subdivided_square_10_00.svg/150px-Subdivided_square_10_00.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Subdivided_square_10_00.svg/200px-Subdivided_square_10_00.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_o10C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Conway_o10C.png/100px-Conway_o10C.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Conway_o10C.png/150px-Conway_o10C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Conway_o10C.png/200px-Conway_o10C.png 2x" data-file-width="366" data-file-height="361" /></a></span><br /><i>o</i><sub>10</sub> = <i>oo</i><sub>2,1</sub><i>o</i><sub>1,2</sub> </td> <td colspan="2"><i>e</i><sub>10</sub> = <i>ee</i><sub>2,1</sub>e<sub>1,2</sub> </td></tr> <tr> <td>1 </td> <td>1 </td> <td>II </td> <td>2 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d1d55e81b3442e9a7775fa603c69f82dd3e9b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&1\\0&2&0\\0&1&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_01_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Subdivided_square_01_01.svg/120px-Subdivided_square_01_01.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Subdivided_square_01_01.svg/150px-Subdivided_square_01_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Subdivided_square_01_01.svg/200px-Subdivided_square_01_01.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_jC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_jC.png/120px-Conway_jC.png" decoding="async" width="100" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Conway_jC.png/150px-Conway_jC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/51/Conway_jC.png 2x" data-file-width="184" data-file-height="192" /></a></span><br /><i>o</i><sub>1,1</sub> = <i>j</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_aC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_aC.png/100px-Conway_aC.png" decoding="async" width="100" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Conway_aC.png/150px-Conway_aC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/87/Conway_aC.png 2x" data-file-width="195" data-file-height="198" /></a></span><br /><i>e</i><sub>1,1</sub> = <i>a</i> </td></tr> <tr> <td>2 </td> <td>2 </td> <td>II </td> <td>8 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&3&1\\0&8&0\\0&4&0\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> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&1\\0&8&0\\0&4&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6109206fdbfa47111d9bf2375fe4a676064a550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&3&1\\0&8&0\\0&4&0\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_02_02.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Subdivided_square_02_02.svg/120px-Subdivided_square_02_02.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Subdivided_square_02_02.svg/150px-Subdivided_square_02_02.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Subdivided_square_02_02.svg/200px-Subdivided_square_02_02.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_daaaC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Conway_daaaC.png/100px-Conway_daaaC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Conway_daaaC.png/150px-Conway_daaaC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Conway_daaaC.png/200px-Conway_daaaC.png 2x" data-file-width="395" data-file-height="406" /></a></span><br /><i>o</i><sub>2,2</sub> = <i>j</i><sup>3</sup> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_aaaC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Conway_aaaC.png/100px-Conway_aaaC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Conway_aaaC.png/150px-Conway_aaaC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Conway_aaaC.png/200px-Conway_aaaC.png 2x" data-file-width="395" data-file-height="406" /></a></span><br /><i>e</i><sub>2,2</sub> = <i>a</i><sup>3</sup> </td></tr> <tr> <td>1 </td> <td>2 </td> <td>III </td> <td>5 </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f17c726d82d518ea3355b897a9268fdaccd877df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&0\\0&5&0\\0&2&1\end{bmatrix}}}" /></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Subdivided_square_01_02.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Subdivided_square_01_02.svg/100px-Subdivided_square_01_02.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Subdivided_square_01_02.svg/150px-Subdivided_square_01_02.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Subdivided_square_01_02.svg/200px-Subdivided_square_01_02.svg.png 2x" data-file-width="182" data-file-height="182" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_pC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/100px-Conway_pC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/150px-Conway_pC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/200px-Conway_pC.png 2x" data-file-width="381" data-file-height="392" /></a></span><br /><i>o</i><sub>1,2</sub> = <i>p</i> </td> <td colspan="2"><span typeof="mw:File"><a href="/wiki/File:Conway_dpC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Conway_dpC.png/100px-Conway_dpC.png" decoding="async" width="100" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Conway_dpC.png/150px-Conway_dpC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Conway_dpC.png/200px-Conway_dpC.png 2x" data-file-width="375" data-file-height="380" /></a></span><br /><i>e</i><sub>1,2</sub> = <i>dp</i> = <i>pd</i> </td> <td><span typeof="mw:File"><a href="/wiki/File:Conway_pC.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/100px-Conway_pC.png" decoding="async" width="100" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/150px-Conway_pC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conway_pC.png/200px-Conway_pC.png 2x" data-file-width="381" data-file-height="392" /></a></span><br /><i>p</i> </td></tr> <tr> <td colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\equiv b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859529d640f85a2c3bf2847f61a842ba3a5753ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a\equiv b}" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\mathrm {mod} \ 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\mathrm {mod} \ 2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a91d2c3cd911d0e7ee35b6e8bedcfc7fe15b7083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.524ex; height:2.843ex;" alt="{\displaystyle \ (\mathrm {mod} \ 2)}" /></span> </td> <td>I, II, or III </td> <td><i>T</i> even </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&{\frac {T}{2}}-1&1\\0&T&0\\0&{\frac {T}{2}}&0\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> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>T</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&{\frac {T}{2}}-1&1\\0&T&0\\0&{\frac {T}{2}}&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3706c2071005c0cdf521abc102ee4a2a3d523b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:16.818ex; height:10.843ex;" alt="{\displaystyle {\begin{bmatrix}1&{\frac {T}{2}}-1&1\\0&T&0\\0&{\frac {T}{2}}&0\end{bmatrix}}}" /></span> </td> <td>... </td> <td colspan="2"><i>o<sub>a,b</sub></i> </td> <td colspan="2"><i>e<sub>a,b</sub></i> </td></tr> <tr> <td colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\not \equiv b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≢</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\not \equiv b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08277b01c0fff9691e6bc25c2ca7663d9059ad07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.327ex; height:2.676ex;" alt="{\displaystyle a\not \equiv b}" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\mathrm {mod} \ 2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\mathrm {mod} \ 2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a91d2c3cd911d0e7ee35b6e8bedcfc7fe15b7083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.524ex; height:2.843ex;" alt="{\displaystyle \ (\mathrm {mod} \ 2)}" /></span> </td> <td>I or III </td> <td><i>T</i> odd </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&{\frac {T-1}{2}}&0\\0&T&0\\0&{\frac {T-1}{2}}&1\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> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>T</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&{\frac {T-1}{2}}&0\\0&T&0\\0&{\frac {T-1}{2}}&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0c71a9a2f880b0837dc69849fbe437910f0482" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:14.916ex; height:11.176ex;" alt="{\displaystyle {\begin{bmatrix}1&{\frac {T-1}{2}}&0\\0&T&0\\0&{\frac {T-1}{2}}&1\end{bmatrix}}}" /></span> </td> <td>... </td> <td><i>o<sub>a,b</sub></i> </td> <td colspan="2"><i>e<sub>a,b</sub></i> </td> <td><i>o<sub>a,b</sub></i> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=14" title="Edit section: Examples"><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/List_of_geodesic_polyhedra_and_Goldberg_polyhedra" title="List of geodesic polyhedra and Goldberg polyhedra">List of geodesic polyhedra and Goldberg polyhedra</a></div> <div class="mw-heading mw-heading3"><h3 id="Archimedean_and_Catalan_solids">Archimedean and Catalan solids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=15" title="Edit section: Archimedean and Catalan solids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conway's original set of operators can create all of the <a href="/wiki/Archimedean_solids" class="mw-redirect" title="Archimedean solids">Archimedean solids</a> and <a href="/wiki/Catalan_solids" class="mw-redirect" title="Catalan solids">Catalan solids</a>, using the <a href="/wiki/Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a> as seeds. (Note that the <i>r</i> operator is not necessary to create both chiral forms.) </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Archimedean</li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_tetrahedron.png" class="mw-file-description" title="Truncated tetrahedron tT"><img alt="Truncated tetrahedron tT" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Truncated_tetrahedron.png/120px-Truncated_tetrahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Truncated_tetrahedron.png/180px-Truncated_tetrahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Truncated_tetrahedron.png/240px-Truncated_tetrahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">Truncated tetrahedron</a><br /><i>tT</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Cuboctahedron.png" class="mw-file-description" title="Cuboctahedron aC = aO = eT"><img alt="Cuboctahedron aC = aO = eT" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Cuboctahedron.png/120px-Cuboctahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Cuboctahedron.png/180px-Cuboctahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Cuboctahedron.png/240px-Cuboctahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Cuboctahedron" title="Cuboctahedron">Cuboctahedron</a><br /><i>aC</i> = <i>aO</i> = <i>eT</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_hexahedron.png" class="mw-file-description" title="Truncated cube tC"><img alt="Truncated cube tC" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Truncated_hexahedron.png/120px-Truncated_hexahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Truncated_hexahedron.png/180px-Truncated_hexahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Truncated_hexahedron.png/240px-Truncated_hexahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_cube" title="Truncated cube">Truncated cube</a><br /><i>tC</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_octahedron.png" class="mw-file-description" title="Truncated octahedron tO = bT"><img alt="Truncated octahedron tO = bT" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Truncated_octahedron.png/120px-Truncated_octahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Truncated_octahedron.png/180px-Truncated_octahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Truncated_octahedron.png/240px-Truncated_octahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">Truncated octahedron</a><br /><i>tO</i> = <i>bT</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Small_rhombicuboctahedron.png" class="mw-file-description" title="Rhombicuboctahedron eC = eO"><img alt="Rhombicuboctahedron eC = eO" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Small_rhombicuboctahedron.png/120px-Small_rhombicuboctahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Small_rhombicuboctahedron.png/180px-Small_rhombicuboctahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Small_rhombicuboctahedron.png/240px-Small_rhombicuboctahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">Rhombicuboctahedron</a><br /><i>eC</i> = <i>eO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Great_rhombicuboctahedron.png" class="mw-file-description" title="truncated cuboctahedron bC = bO"><img alt="truncated cuboctahedron bC = bO" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Great_rhombicuboctahedron.png/120px-Great_rhombicuboctahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Great_rhombicuboctahedron.png/180px-Great_rhombicuboctahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Great_rhombicuboctahedron.png/240px-Great_rhombicuboctahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a><br /><i>bC</i> = <i>bO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Snub_hexahedron.png" class="mw-file-description" title="snub cube sC = sO"><img alt="snub cube sC = sO" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Snub_hexahedron.png/120px-Snub_hexahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Snub_hexahedron.png/180px-Snub_hexahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Snub_hexahedron.png/240px-Snub_hexahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Snub_cube" title="Snub cube">snub cube</a><br /><i>sC</i> = <i>sO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Icosidodecahedron.png" class="mw-file-description" title="icosidodecahedron aD = aI"><img alt="icosidodecahedron aD = aI" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/120px-Icosidodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/180px-Icosidodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Icosidodecahedron.png/240px-Icosidodecahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a><br /><i>aD</i> = <i>aI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_dodecahedron.png" class="mw-file-description" title="truncated dodecahedron tD"><img alt="truncated dodecahedron tD" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Truncated_dodecahedron.png/120px-Truncated_dodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Truncated_dodecahedron.png/250px-Truncated_dodecahedron.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a><br /><i>tD</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_icosahedron.png" class="mw-file-description" title="truncated icosahedron tI"><img alt="truncated icosahedron tI" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Truncated_icosahedron.png/120px-Truncated_icosahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Truncated_icosahedron.png/180px-Truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Truncated_icosahedron.png/240px-Truncated_icosahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a><br /><i>tI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Small_rhombicosidodecahedron.png" class="mw-file-description" title="rhombicosidodecahedron eD = eI"><img alt="rhombicosidodecahedron eD = eI" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Small_rhombicosidodecahedron.png/120px-Small_rhombicosidodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Small_rhombicosidodecahedron.png/180px-Small_rhombicosidodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Small_rhombicosidodecahedron.png/240px-Small_rhombicosidodecahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a><br /><i>eD</i> = <i>eI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Great_rhombicosidodecahedron.png" class="mw-file-description" title="truncated icosidodecahedron bD = bI"><img alt="truncated icosidodecahedron bD = bI" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Great_rhombicosidodecahedron.png/120px-Great_rhombicosidodecahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Great_rhombicosidodecahedron.png/180px-Great_rhombicosidodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Great_rhombicosidodecahedron.png/240px-Great_rhombicosidodecahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a><br /><i>bD</i> = <i>bI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Snub_dodecahedron_ccw.png" class="mw-file-description" title="snub dodecahedron sD = sI"><img alt="snub dodecahedron sD = sI" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Snub_dodecahedron_ccw.png/120px-Snub_dodecahedron_ccw.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Snub_dodecahedron_ccw.png/180px-Snub_dodecahedron_ccw.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Snub_dodecahedron_ccw.png/240px-Snub_dodecahedron_ccw.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a><br /><i>sD</i> = <i>sI</i></div> </li> </ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Catalan</li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Triakistetrahedron.jpg" class="mw-file-description" title="Triakis tetrahedron kT"><img alt="Triakis tetrahedron kT" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Triakistetrahedron.jpg/120px-Triakistetrahedron.jpg" decoding="async" width="107" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Triakistetrahedron.jpg/250px-Triakistetrahedron.jpg 1.5x" data-file-width="681" data-file-height="766" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">Triakis tetrahedron</a><br /><i>kT</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Rhombicdodecahedron.jpg" class="mw-file-description" title="Rhombic dodecahedron jC = jO = oT"><img alt="Rhombic dodecahedron jC = jO = oT" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/120px-Rhombicdodecahedron.jpg" decoding="async" width="120" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/180px-Rhombicdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/240px-Rhombicdodecahedron.jpg 2x" data-file-width="849" data-file-height="754" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Rhombic dodecahedron</a><br /><i>jC</i> = <i>jO</i> = <i>oT</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Triakisoctahedron.jpg" class="mw-file-description" title="Triakis octahedron kO"><img alt="Triakis octahedron kO" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Triakisoctahedron.jpg/120px-Triakisoctahedron.jpg" decoding="async" width="109" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Triakisoctahedron.jpg/250px-Triakisoctahedron.jpg 1.5x" data-file-width="679" data-file-height="745" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Triakis_octahedron" title="Triakis octahedron">Triakis octahedron</a><br /><i>kO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Tetrakishexahedron.jpg" class="mw-file-description" title="Tetrakis hexahedron kC = mT"><img alt="Tetrakis hexahedron kC = mT" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/120px-Tetrakishexahedron.jpg" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/180px-Tetrakishexahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Tetrakishexahedron.jpg/240px-Tetrakishexahedron.jpg 2x" data-file-width="767" data-file-height="737" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">Tetrakis hexahedron</a><br /><i>kC</i> = <i>mT</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Deltoidalicositetrahedron.jpg" class="mw-file-description" title="Deltoidal icositetrahedron oC = oO"><img alt="Deltoidal icositetrahedron oC = oO" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/120px-Deltoidalicositetrahedron.jpg" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Deltoidalicositetrahedron.jpg/250px-Deltoidalicositetrahedron.jpg 1.5x" data-file-width="845" data-file-height="837" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">Deltoidal icositetrahedron</a><br /><i>oC</i> = <i>oO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Disdyakisdodecahedron.jpg" class="mw-file-description" title="Disdyakis dodecahedron mC = mO"><img alt="Disdyakis dodecahedron mC = mO" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Disdyakisdodecahedron.jpg/114px-Disdyakisdodecahedron.jpg" decoding="async" width="114" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Disdyakisdodecahedron.jpg/172px-Disdyakisdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Disdyakisdodecahedron.jpg/229px-Disdyakisdodecahedron.jpg 2x" data-file-width="765" data-file-height="802" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">Disdyakis dodecahedron</a><br /><i>mC</i> = <i>mO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pentagonalicositetrahedronccw.jpg" class="mw-file-description" title="Pentagonal icositetrahedron gC = gO"><img alt="Pentagonal icositetrahedron gC = gO" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Pentagonalicositetrahedronccw.jpg/120px-Pentagonalicositetrahedronccw.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Pentagonalicositetrahedronccw.jpg/250px-Pentagonalicositetrahedronccw.jpg 1.5x" data-file-width="829" data-file-height="813" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">Pentagonal icositetrahedron</a><br /><i>gC</i> = <i>gO</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Rhombictriacontahedron.svg" class="mw-file-description" title="Rhombic triacontahedron jD = jI"><img alt="Rhombic triacontahedron jD = jI" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Rhombictriacontahedron.svg/108px-Rhombictriacontahedron.svg.png" decoding="async" width="108" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Rhombictriacontahedron.svg/163px-Rhombictriacontahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Rhombictriacontahedron.svg/217px-Rhombictriacontahedron.svg.png 2x" data-file-width="560" data-file-height="620" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">Rhombic triacontahedron</a><br /><i>jD</i> = <i>jI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Triakisicosahedron.jpg" class="mw-file-description" title="Triakis icosahedron kI"><img alt="Triakis icosahedron kI" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/116px-Triakisicosahedron.jpg" decoding="async" width="116" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/174px-Triakisicosahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Triakisicosahedron.jpg/232px-Triakisicosahedron.jpg 2x" data-file-width="819" data-file-height="849" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">Triakis icosahedron</a><br /><i>kI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pentakisdodecahedron.jpg" class="mw-file-description" title="Pentakis dodecahedron kD"><img alt="Pentakis dodecahedron kD" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/120px-Pentakisdodecahedron.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/180px-Pentakisdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/240px-Pentakisdodecahedron.jpg 2x" data-file-width="844" data-file-height="843" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">Pentakis dodecahedron</a><br /><i>kD</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Deltoidalhexecontahedron.jpg" class="mw-file-description" title="Deltoidal hexecontahedron oD = oI"><img alt="Deltoidal hexecontahedron oD = oI" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/120px-Deltoidalhexecontahedron.jpg" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/180px-Deltoidalhexecontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Deltoidalhexecontahedron.jpg/240px-Deltoidalhexecontahedron.jpg 2x" data-file-width="854" data-file-height="843" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">Deltoidal hexecontahedron</a><br /><i>oD</i> = <i>oI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Disdyakistriacontahedron.jpg" class="mw-file-description" title="Disdyakis triacontahedron mD = mI"><img alt="Disdyakis triacontahedron mD = mI" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/115px-Disdyakistriacontahedron.jpg" decoding="async" width="115" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/173px-Disdyakistriacontahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Disdyakistriacontahedron.jpg/230px-Disdyakistriacontahedron.jpg 2x" data-file-width="812" data-file-height="847" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">Disdyakis triacontahedron</a><br /><i>mD</i> = <i>mI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pentagonalhexecontahedronccw.jpg" class="mw-file-description" title="Pentagonal hexecontahedron gD = gI"><img alt="Pentagonal hexecontahedron gD = gI" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Pentagonalhexecontahedronccw.jpg/120px-Pentagonalhexecontahedronccw.jpg" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Pentagonalhexecontahedronccw.jpg/250px-Pentagonalhexecontahedronccw.jpg 1.5x" data-file-width="834" data-file-height="851" /></a></span></div> <div class="gallerytext"> <a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">Pentagonal hexecontahedron</a><br /><i>gD</i> = <i>gI</i></div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Composite_operators">Composite operators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=16" title="Edit section: Composite operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a>, <i>tI</i>, can be used as a seed to create some more visually-pleasing polyhedra, although these are neither <a href="/wiki/Vertex_transitive" class="mw-redirect" title="Vertex transitive">vertex</a> nor <a href="/wiki/Face-transitive" class="mw-redirect" title="Face-transitive">face-transitive</a>. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-53-t12.svg" class="mw-file-description" title="tI"><img alt="tI" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Uniform_polyhedron-53-t12.svg/120px-Uniform_polyhedron-53-t12.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Uniform_polyhedron-53-t12.svg/180px-Uniform_polyhedron-53-t12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Uniform_polyhedron-53-t12.svg/240px-Uniform_polyhedron-53-t12.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span></div> <div class="gallerytext"><i>tI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Rectified_truncated_icosahedron.png" class="mw-file-description" title="atI"><img alt="atI" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Rectified_truncated_icosahedron.png/120px-Rectified_truncated_icosahedron.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Rectified_truncated_icosahedron.png/180px-Rectified_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Rectified_truncated_icosahedron.png/240px-Rectified_truncated_icosahedron.png 2x" data-file-width="418" data-file-height="412" /></a></span></div> <div class="gallerytext"><i>atI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_truncated_icosahedron.png" class="mw-file-description" title="ttI"><img alt="ttI" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Truncated_truncated_icosahedron.png/120px-Truncated_truncated_icosahedron.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Truncated_truncated_icosahedron.png/180px-Truncated_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Truncated_truncated_icosahedron.png/240px-Truncated_truncated_icosahedron.png 2x" data-file-width="404" data-file-height="398" /></a></span></div> <div class="gallerytext"><i>ttI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Conway_polyhedron_Dk6k5tI.png" class="mw-file-description" title="ztI = ttD"><img alt="ztI = ttD" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Conway_polyhedron_Dk6k5tI.png/120px-Conway_polyhedron_Dk6k5tI.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Conway_polyhedron_Dk6k5tI.png/180px-Conway_polyhedron_Dk6k5tI.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Conway_polyhedron_Dk6k5tI.png/240px-Conway_polyhedron_Dk6k5tI.png 2x" data-file-width="900" data-file-height="883" /></a></span></div> <div class="gallerytext"><i>ztI</i> = <i>ttD</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Expanded_truncated_icosahedron.png" class="mw-file-description" title="etI"><img alt="etI" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Expanded_truncated_icosahedron.png/120px-Expanded_truncated_icosahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Expanded_truncated_icosahedron.png/180px-Expanded_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Expanded_truncated_icosahedron.png/240px-Expanded_truncated_icosahedron.png 2x" data-file-width="401" data-file-height="401" /></a></span></div> <div class="gallerytext"><i>etI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_rectified_truncated_icosahedron.png" class="mw-file-description" title="btI"><img alt="btI" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Truncated_rectified_truncated_icosahedron.png/120px-Truncated_rectified_truncated_icosahedron.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Truncated_rectified_truncated_icosahedron.png/180px-Truncated_rectified_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Truncated_rectified_truncated_icosahedron.png/240px-Truncated_rectified_truncated_icosahedron.png 2x" data-file-width="415" data-file-height="407" /></a></span></div> <div class="gallerytext"><i>btI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Snub_rectified_truncated_icosahedron.png" class="mw-file-description" title="stI"><img alt="stI" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Snub_rectified_truncated_icosahedron.png/119px-Snub_rectified_truncated_icosahedron.png" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Snub_rectified_truncated_icosahedron.png/179px-Snub_rectified_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Snub_rectified_truncated_icosahedron.png/239px-Snub_rectified_truncated_icosahedron.png 2x" data-file-width="391" data-file-height="393" /></a></span></div> <div class="gallerytext"><i>stI</i></div> </li> </ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Duals</li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pentakisdodecahedron.jpg" class="mw-file-description" title="dtI = nI = kD"><img alt="dtI = nI = kD" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/120px-Pentakisdodecahedron.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/180px-Pentakisdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Pentakisdodecahedron.jpg/240px-Pentakisdodecahedron.jpg 2x" data-file-width="844" data-file-height="843" /></a></span></div> <div class="gallerytext"><i>dtI</i> = <i>nI</i> = <i>kD</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Joined_truncated_icosahedron.png" class="mw-file-description" title="jtI"><img alt="jtI" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Joined_truncated_icosahedron.png/120px-Joined_truncated_icosahedron.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Joined_truncated_icosahedron.png/180px-Joined_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Joined_truncated_icosahedron.png/240px-Joined_truncated_icosahedron.png 2x" data-file-width="407" data-file-height="400" /></a></span></div> <div class="gallerytext"><i>jtI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Kissed_kissed_dodecahedron.png" class="mw-file-description" title="ntI = kkD"><img alt="ntI = kkD" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Kissed_kissed_dodecahedron.png/120px-Kissed_kissed_dodecahedron.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Kissed_kissed_dodecahedron.png/180px-Kissed_kissed_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Kissed_kissed_dodecahedron.png/240px-Kissed_kissed_dodecahedron.png 2x" data-file-width="393" data-file-height="387" /></a></span></div> <div class="gallerytext"><i>ntI</i> = <i>kkD</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Conway_polyhedron_K6k5tI.png" class="mw-file-description" title="ktI"><img alt="ktI" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Conway_polyhedron_K6k5tI.png/120px-Conway_polyhedron_K6k5tI.png" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Conway_polyhedron_K6k5tI.png/250px-Conway_polyhedron_K6k5tI.png 1.5x" data-file-width="913" data-file-height="903" /></a></span></div> <div class="gallerytext"><i>ktI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Ortho_truncated_icosahedron.png" class="mw-file-description" title="otI"><img alt="otI" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ortho_truncated_icosahedron.png/120px-Ortho_truncated_icosahedron.png" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ortho_truncated_icosahedron.png/180px-Ortho_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Ortho_truncated_icosahedron.png/240px-Ortho_truncated_icosahedron.png 2x" data-file-width="397" data-file-height="394" /></a></span></div> <div class="gallerytext"><i>otI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Meta_truncated_icosahedron.png" class="mw-file-description" title="mtI"><img alt="mtI" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Meta_truncated_icosahedron.png/120px-Meta_truncated_icosahedron.png" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Meta_truncated_icosahedron.png/180px-Meta_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Meta_truncated_icosahedron.png/240px-Meta_truncated_icosahedron.png 2x" data-file-width="416" data-file-height="403" /></a></span></div> <div class="gallerytext"><i>mtI</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Gyro_truncated_icosahedron.png" class="mw-file-description" title="gtI"><img alt="gtI" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Gyro_truncated_icosahedron.png/120px-Gyro_truncated_icosahedron.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Gyro_truncated_icosahedron.png/180px-Gyro_truncated_icosahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Gyro_truncated_icosahedron.png/240px-Gyro_truncated_icosahedron.png 2x" data-file-width="406" data-file-height="406" /></a></span></div> <div class="gallerytext"><i>gtI</i></div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="On_the_plane">On the plane</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=17" title="Edit section: On the plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each of the <a href="/wiki/List_of_convex_uniform_tilings" class="mw-redirect" title="List of convex uniform tilings">convex uniform tilings</a> and their duals can be created by applying Conway operators to the <a href="/wiki/Regular_tilings" class="mw-redirect" title="Regular tilings">regular tilings</a> <i>Q</i>, <i>H</i>, and <i>Δ</i>. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n5.svg" class="mw-file-description" title="Square tiling Q = dQ = aQ = eQ = jQ = oQ"><img alt="Square tiling Q = dQ = aQ = eQ = jQ = oQ" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/1-uniform_n5.svg/120px-1-uniform_n5.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/1-uniform_n5.svg/250px-1-uniform_n5.svg.png 1.5x" data-file-width="384" data-file-height="384" /></a></span></div> <div class="gallerytext"><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a><br /><i>Q</i> = <i>dQ</i> = <i>aQ</i> = <i>eQ</i><br />= <i>jQ</i> = <i>oQ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n2.svg" class="mw-file-description" title="Truncated square tiling tQ = bQ"><img alt="Truncated square tiling tQ = bQ" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/1-uniform_n2.svg/120px-1-uniform_n2.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/1-uniform_n2.svg/180px-1-uniform_n2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/1-uniform_n2.svg/240px-1-uniform_n2.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Truncated_square_tiling" title="Truncated square tiling">Truncated square tiling</a><br /><i>tQ</i> = <i>bQ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_2_dual.svg" class="mw-file-description" title="Tetrakis square tiling kQ = mQ"><img alt="Tetrakis square tiling kQ = mQ" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/1-uniform_2_dual.svg/120px-1-uniform_2_dual.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/1-uniform_2_dual.svg/180px-1-uniform_2_dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/1-uniform_2_dual.svg/240px-1-uniform_2_dual.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Tetrakis_square_tiling" title="Tetrakis square tiling">Tetrakis square tiling</a><br /><i>kQ</i> = <i>mQ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n9.svg" class="mw-file-description" title="Snub square tiling sQ"><img alt="Snub square tiling sQ" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/1-uniform_n9.svg/120px-1-uniform_n9.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/1-uniform_n9.svg/250px-1-uniform_n9.svg.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Snub_square_tiling" title="Snub square tiling">Snub square tiling</a><br /><i>sQ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_9_dual.svg" class="mw-file-description" title="Cairo pentagonal tiling gQ"><img alt="Cairo pentagonal tiling gQ" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/1-uniform_9_dual.svg/120px-1-uniform_9_dual.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/1-uniform_9_dual.svg/180px-1-uniform_9_dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/1-uniform_9_dual.svg/240px-1-uniform_9_dual.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Cairo_pentagonal_tiling" title="Cairo pentagonal tiling">Cairo pentagonal tiling</a><br /><i>gQ</i></div> </li> </ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n1.svg" class="mw-file-description" title="Hexagonal tiling H = dΔ = tΔ"><img alt="Hexagonal tiling H = dΔ = tΔ" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/1-uniform_n1.svg/120px-1-uniform_n1.svg.png" decoding="async" width="120" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/1-uniform_n1.svg/250px-1-uniform_n1.svg.png 1.5x" data-file-width="1183" data-file-height="1047" /></a></span></div> <div class="gallerytext"><a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">Hexagonal tiling</a><br /><i>H</i> = <i>dΔ</i> = <i>tΔ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n7.svg" class="mw-file-description" title="Trihexagonal tiling aH = aΔ"><img alt="Trihexagonal tiling aH = aΔ" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/1-uniform_n7.svg/120px-1-uniform_n7.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/1-uniform_n7.svg/180px-1-uniform_n7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/1-uniform_n7.svg/240px-1-uniform_n7.svg.png 2x" data-file-width="1000" data-file-height="979" /></a></span></div> <div class="gallerytext"><a href="/wiki/Trihexagonal_tiling" title="Trihexagonal tiling">Trihexagonal tiling</a><br /><i>aH</i> = <i>aΔ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n4.svg" class="mw-file-description" title="Truncated hexagonal tiling tH"><img alt="Truncated hexagonal tiling tH" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/1-uniform_n4.svg/120px-1-uniform_n4.svg.png" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/1-uniform_n4.svg/250px-1-uniform_n4.svg.png 1.5x" data-file-width="995" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Truncated_hexagonal_tiling" title="Truncated hexagonal tiling">Truncated hexagonal tiling</a><br /><i>tH</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n6.svg" class="mw-file-description" title="Rhombitrihexagonal tiling eH = eΔ"><img alt="Rhombitrihexagonal tiling eH = eΔ" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/1-uniform_n6.svg/120px-1-uniform_n6.svg.png" decoding="async" width="119" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/1-uniform_n6.svg/250px-1-uniform_n6.svg.png 1.5x" data-file-width="991" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Rhombitrihexagonal_tiling" title="Rhombitrihexagonal tiling">Rhombitrihexagonal tiling</a><br /><i>eH</i> = <i>eΔ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n3.svg" class="mw-file-description" title="Truncated trihexagonal tiling bH = bΔ"><img alt="Truncated trihexagonal tiling bH = bΔ" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/1-uniform_n3.svg/120px-1-uniform_n3.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/1-uniform_n3.svg/180px-1-uniform_n3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/1-uniform_n3.svg/240px-1-uniform_n3.svg.png 2x" data-file-width="1000" data-file-height="997" /></a></span></div> <div class="gallerytext"><a href="/wiki/Truncated_trihexagonal_tiling" title="Truncated trihexagonal tiling">Truncated trihexagonal tiling</a><br /><i>bH</i> = <i>bΔ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_n10.svg" class="mw-file-description" title="Snub trihexagonal tiling sH = sΔ"><img alt="Snub trihexagonal tiling sH = sΔ" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/1-uniform_n10.svg/120px-1-uniform_n10.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/1-uniform_n10.svg/250px-1-uniform_n10.svg.png 1.5x" data-file-width="1000" data-file-height="979" /></a></span></div> <div class="gallerytext"><a href="/wiki/Snub_trihexagonal_tiling" title="Snub trihexagonal tiling">Snub trihexagonal tiling</a><br /><i>sH</i> = <i>sΔ</i></div> </li> </ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_1_dual.svg" class="mw-file-description" title="Triangle tiling Δ = dH = kH"><img alt="Triangle tiling Δ = dH = kH" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/1-uniform_1_dual.svg/120px-1-uniform_1_dual.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/1-uniform_1_dual.svg/250px-1-uniform_1_dual.svg.png 1.5x" data-file-width="1000" data-file-height="977" /></a></span></div> <div class="gallerytext"><a href="/wiki/Triangle_tiling" class="mw-redirect" title="Triangle tiling">Triangle tiling</a><br /><i>Δ</i> = <i>dH</i> = <i>kH</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_7_dual.svg" class="mw-file-description" title="Rhombille tiling jΔ = jH"><img alt="Rhombille tiling jΔ = jH" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/1-uniform_7_dual.svg/120px-1-uniform_7_dual.svg.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/1-uniform_7_dual.svg/180px-1-uniform_7_dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/1-uniform_7_dual.svg/240px-1-uniform_7_dual.svg.png 2x" data-file-width="1000" data-file-height="962" /></a></span></div> <div class="gallerytext"><a href="/wiki/Rhombille_tiling" title="Rhombille tiling">Rhombille tiling</a><br /><i>jΔ</i> = <i>jH</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_4_dual.svg" class="mw-file-description" title="Triakis triangular tiling kΔ"><img alt="Triakis triangular tiling kΔ" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/1-uniform_4_dual.svg/110px-1-uniform_4_dual.svg.png" decoding="async" width="110" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/1-uniform_4_dual.svg/165px-1-uniform_4_dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/1-uniform_4_dual.svg/220px-1-uniform_4_dual.svg.png 2x" data-file-width="917" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Triakis_triangular_tiling" class="mw-redirect" title="Triakis triangular tiling">Triakis triangular tiling</a><br /><i>kΔ</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_6_dual.svg" class="mw-file-description" title="Deltoidal trihexagonal tiling oΔ = oH"><img alt="Deltoidal trihexagonal tiling oΔ = oH" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/1-uniform_6_dual.svg/116px-1-uniform_6_dual.svg.png" decoding="async" width="116" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/1-uniform_6_dual.svg/174px-1-uniform_6_dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/1-uniform_6_dual.svg/232px-1-uniform_6_dual.svg.png 2x" data-file-width="968" data-file-height="1000" /></a></span></div> <div class="gallerytext"><a href="/wiki/Deltoidal_trihexagonal_tiling" class="mw-redirect" title="Deltoidal trihexagonal tiling">Deltoidal trihexagonal tiling</a><br /><i>oΔ</i> = <i>oH</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_3_dual.svg" class="mw-file-description" title="Kisrhombille tiling mΔ = mH"><img alt="Kisrhombille tiling mΔ = mH" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/1-uniform_3_dual.svg/120px-1-uniform_3_dual.svg.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/1-uniform_3_dual.svg/250px-1-uniform_3_dual.svg.png 1.5x" data-file-width="1000" data-file-height="981" /></a></span></div> <div class="gallerytext"><a href="/wiki/Kisrhombille_tiling" class="mw-redirect" title="Kisrhombille tiling">Kisrhombille tiling</a><br /><i>mΔ</i> = <i>mH</i></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:1-uniform_10_dual.svg" class="mw-file-description" title="Floret pentagonal tiling gΔ = gH"><img alt="Floret pentagonal tiling gΔ = gH" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/1-uniform_10_dual.svg/120px-1-uniform_10_dual.svg.png" decoding="async" width="120" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/1-uniform_10_dual.svg/180px-1-uniform_10_dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/1-uniform_10_dual.svg/240px-1-uniform_10_dual.svg.png 2x" data-file-width="1000" data-file-height="919" /></a></span></div> <div class="gallerytext"><a href="/wiki/Floret_pentagonal_tiling" class="mw-redirect" title="Floret pentagonal tiling">Floret pentagonal tiling</a><br /><i>gΔ</i> = <i>gH</i></div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="On_a_torus">On a torus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=18" title="Edit section: On a torus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conway operators can also be applied to <a href="/wiki/Toroidal_polyhedra" class="mw-redirect" title="Toroidal polyhedra">toroidal polyhedra</a> and polyhedra with multiple holes. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Toroidal_monohedron.png" class="mw-file-description" title="A 1x1 regular square torus, {4,4}1,0"><img alt="A 1x1 regular square torus, {4,4}1,0" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Toroidal_monohedron.png/120px-Toroidal_monohedron.png" decoding="async" width="120" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Toroidal_monohedron.png/180px-Toroidal_monohedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Toroidal_monohedron.png/240px-Toroidal_monohedron.png 2x" data-file-width="1409" data-file-height="896" /></a></span></div> <div class="gallerytext">A 1x1 regular square torus, {4,4}<sub>1,0</sub></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Torus_map_4x4.png" class="mw-file-description" title="A regular 4x4 square torus, {4,4}4,0"><img alt="A regular 4x4 square torus, {4,4}4,0" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Torus_map_4x4.png/120px-Torus_map_4x4.png" decoding="async" width="120" height="84" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Torus_map_4x4.png/180px-Torus_map_4x4.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Torus_map_4x4.png/240px-Torus_map_4x4.png 2x" data-file-width="1195" data-file-height="839" /></a></span></div> <div class="gallerytext">A regular 4x4 square torus, {4,4}<sub>4,0</sub></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:First_truncated_square_tiling_on_torus24x12.png" class="mw-file-description" title="tQ24×12 projected to torus"><img alt="tQ24×12 projected to torus" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/First_truncated_square_tiling_on_torus24x12.png/120px-First_truncated_square_tiling_on_torus24x12.png" decoding="async" width="120" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/First_truncated_square_tiling_on_torus24x12.png/180px-First_truncated_square_tiling_on_torus24x12.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/First_truncated_square_tiling_on_torus24x12.png/240px-First_truncated_square_tiling_on_torus24x12.png 2x" data-file-width="1390" data-file-height="1100" /></a></span></div> <div class="gallerytext">tQ24×12 projected to torus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_square_tiling_on_torus24x12.png" class="mw-file-description" title="taQ24×12 projected to torus"><img alt="taQ24×12 projected to torus" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Truncated_square_tiling_on_torus24x12.png/120px-Truncated_square_tiling_on_torus24x12.png" decoding="async" width="120" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Truncated_square_tiling_on_torus24x12.png/180px-Truncated_square_tiling_on_torus24x12.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Truncated_square_tiling_on_torus24x12.png/240px-Truncated_square_tiling_on_torus24x12.png 2x" data-file-width="1498" data-file-height="1100" /></a></span></div> <div class="gallerytext">taQ24×12 projected to torus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Conway_torus_ActQ24x8.png" class="mw-file-description" title="actQ24×8 projected to torus"><img alt="actQ24×8 projected to torus" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Conway_torus_ActQ24x8.png/120px-Conway_torus_ActQ24x8.png" decoding="async" width="120" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Conway_torus_ActQ24x8.png/180px-Conway_torus_ActQ24x8.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Conway_torus_ActQ24x8.png/240px-Conway_torus_ActQ24x8.png 2x" data-file-width="1610" data-file-height="1180" /></a></span></div> <div class="gallerytext">actQ24×8 projected to torus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_hexagonal_tiling_torus24x12.png" class="mw-file-description" title="tH24×12 projected to torus"><img alt="tH24×12 projected to torus" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Truncated_hexagonal_tiling_torus24x12.png/120px-Truncated_hexagonal_tiling_torus24x12.png" decoding="async" width="120" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Truncated_hexagonal_tiling_torus24x12.png/180px-Truncated_hexagonal_tiling_torus24x12.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Truncated_hexagonal_tiling_torus24x12.png/240px-Truncated_hexagonal_tiling_torus24x12.png 2x" data-file-width="1508" data-file-height="1100" /></a></span></div> <div class="gallerytext">tH24×12 projected to torus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Truncated_trihexagonal_tiling_on_torus24x8.png" class="mw-file-description" title="taH24×8 projected to torus"><img alt="taH24×8 projected to torus" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Truncated_trihexagonal_tiling_on_torus24x8.png/120px-Truncated_trihexagonal_tiling_on_torus24x8.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Truncated_trihexagonal_tiling_on_torus24x8.png/180px-Truncated_trihexagonal_tiling_on_torus24x8.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Truncated_trihexagonal_tiling_on_torus24x8.png/240px-Truncated_trihexagonal_tiling_on_torus24x8.png 2x" data-file-width="1658" data-file-height="1330" /></a></span></div> <div class="gallerytext">taH24×8 projected to torus</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Conway_torus_kH24-12.png" class="mw-file-description" title="kH24×12 projected to torus"><img alt="kH24×12 projected to torus" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Conway_torus_kH24-12.png/120px-Conway_torus_kH24-12.png" decoding="async" width="120" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Conway_torus_kH24-12.png/250px-Conway_torus_kH24-12.png 1.5x" data-file-width="1723" data-file-height="1400" /></a></span></div> <div class="gallerytext">kH24×12 projected to torus</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Conway_polyhedra" class="extiw" title="commons:Conway polyhedra"><span style="font-style:italic; font-weight:bold;">Conway polyhedra</span></a>.</div></div> </div> <ul><li><a href="/wiki/Symmetrohedron" title="Symmetrohedron">Symmetrohedron</a></li> <li><a href="/wiki/Zonohedron" title="Zonohedron">Zonohedron</a></li> <li><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-SoT-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-SoT_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFConwayBurgielGoodman-Strauss2008" class="citation book cs1"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John</a>; Burgiel, Heidi; <a href="/wiki/Chaim_Goodman-Strauss" title="Chaim Goodman-Strauss">Goodman-Strauss, Chaim</a> (2008). "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings". <a href="/wiki/The_Symmetries_of_Things" title="The Symmetries of Things"><i>The Symmetries of Things</i></a>. AK Peters. p. 288. <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"><bdi>978-1-56881-220-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+21%3A+Naming+Archimedean+and+Catalan+polyhedra+and+tilings&rft.btitle=The+Symmetries+of+Things&rft.pages=288&rft.pub=AK+Peters&rft.date=2008&rft.isbn=978-1-56881-220-5&rft.aulast=Conway&rft.aufirst=John&rft.au=Burgiel%2C+Heidi&rft.au=Goodman-Strauss%2C+Chaim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConway+polyhedron+notation" class="Z3988"></span></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"><span class="citation mathworld" id="Reference-Mathworld-Conway_Polyhedron_Notation"><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. 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D.; Shtogrin, M. I. (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HLi4CQAAQBAJ&q=goldberg-coxeter&pg=PA130">"Goldberg–Coxeter Construction and Parameterization"</a>. <i>Geometric Structure of Chemistry-Relevant Graphs: Zigzags and Central Circuits</i>. Springer. pp. <span class="nowrap">131–</span>148. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788132224495" title="Special:BookSources/9788132224495"><bdi>9788132224495</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Goldberg%E2%80%93Coxeter+Construction+and+Parameterization&rft.btitle=Geometric+Structure+of+Chemistry-Relevant+Graphs%3A+Zigzags+and+Central+Circuits&rft.pages=%3Cspan+class%3D%22nowrap%22%3E131-%3C%2Fspan%3E148&rft.pub=Springer&rft.date=2015&rft.isbn=9788132224495&rft.au=Deza%2C+M.-M.&rft.au=Sikiri%C4%87%2C+M.+D.&rft.au=Shtogrin%2C+M.+I.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHLi4CQAAQBAJ%26q%3Dgoldberg-coxeter%26pg%3DPA130&rfr_id=info%3Asid%2Fen.wikipedia.org%3AConway+polyhedron+notation" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Conway_polyhedron_notation&action=edit&section=21" 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="https://levskaya.github.io/polyhedronisme/">polyHédronisme</a>: generates polyhedra in HTML5 canvas, taking Conway notation as input</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl 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navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Convex_polyhedron_navigator" title="Template:Convex polyhedron navigator"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Convex_polyhedron_navigator" title="Template talk:Convex polyhedron navigator"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Convex_polyhedron_navigator" title="Special:EditPage/Template:Convex polyhedron navigator"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Convex_polyhedra686" style="font-size:114%;margin:0 4em">Convex <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> <span class="nobold">(<a href="/wiki/Regular_polyhedron" title="Regular polyhedron">regular</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedron#Regular_tetrahedron" title="Tetrahedron">tetrahedron</a></li> <li><a href="/wiki/Cube" title="Cube">cube</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">octahedron</a></li> <li><a href="/wiki/Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a></li> <li><a href="/wiki/Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Archimedean_solid" title="Archimedean solid">Archimedean solids</a><br /><span class="nobold">(<a href="/wiki/Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li> <li><a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li> <li><a href="/wiki/Truncated_cube" title="Truncated cube">truncated cube</a></li> <li><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li> <li><a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li> <li><a href="/wiki/Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li> <li><a href="/wiki/Snub_cube" title="Snub cube">snub cube</a></li> <li><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li> <li><a href="/wiki/Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li> <li><a href="/wiki/Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li> <li><a href="/wiki/Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li> <li><a href="/wiki/Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li> <li><a href="/wiki/Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Catalan_solid" title="Catalan solid">Catalan solids</a><br /><span class="nobold">(duals of Archimedean)</span></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li> <li><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li> <li><a href="/wiki/Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li> <li><a href="/wiki/Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li> <li><a href="/wiki/Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li> <li><a href="/wiki/Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li> <li><a href="/wiki/Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li> <li><a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li> <li><a href="/wiki/Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li> <li><a href="/wiki/Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li> <li><a href="/wiki/Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li> <li><a href="/wiki/Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li> <li><a href="/wiki/Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Dihedron" title="Dihedron">dihedron</a></i></li> <li><i><a href="/wiki/Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a></li> <li><a href="/wiki/Antiprism" title="Antiprism">antiprisms</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bipyramid" title="Bipyramid">bipyramids</a></li> <li><a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedra</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral others</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></li> <li><a href="/wiki/Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedra</a></li> <li><a href="/wiki/Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramid</a></li> <li><a href="/wiki/Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li> <li><a href="/wiki/Bicupola_(geometry)" class="mw-redirect" title="Bicupola (geometry)">bicupola</a></li> <li><a href="/wiki/Frustum" title="Frustum">frustum</a></li> <li><a href="/wiki/Bifrustum" title="Bifrustum">bifrustum</a></li> <li><a href="/wiki/Rotunda_(geometry)" title="Rotunda (geometry)">rotunda</a></li> <li><a href="/wiki/Birotunda" title="Birotunda">birotunda</a></li> <li><a href="/wiki/Prismatoid" title="Prismatoid">prismatoid</a></li> <li><a href="/wiki/Scutoid" title="Scutoid">scutoid</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Degenerate polyhedra are in <i>italics</i>.</div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: 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Conway polyhedron notation - Wikipedia