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id="toc-Higher_degree_rectifications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher_degree_rectifications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Higher degree rectifications</span> </div> </a> <ul id="toc-Higher_degree_rectifications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_birectification_as_a_final_truncation_to_a_face" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example_of_birectification_as_a_final_truncation_to_a_face"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Example of birectification as a final truncation to a face</span> </div> </a> <ul id="toc-Example_of_birectification_as_a_final_truncation_to_a_face-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_polygons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In polygons</span> </div> </a> <ul id="toc-In_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_polyhedra_and_plane_tilings" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_polyhedra_and_plane_tilings"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In polyhedra and plane tilings</span> </div> </a> <button aria-controls="toc-In_polyhedra_and_plane_tilings-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In polyhedra and plane tilings subsection</span> </button> <ul id="toc-In_polyhedra_and_plane_tilings-sublist" class="vector-toc-list"> <li id="toc-In_nonregular_polyhedra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_nonregular_polyhedra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>In nonregular polyhedra</span> </div> </a> <ul id="toc-In_nonregular_polyhedra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_4-polytopes_and_3D_honeycomb_tessellations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_4-polytopes_and_3D_honeycomb_tessellations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In 4-polytopes and 3D honeycomb tessellations</span> </div> </a> <ul id="toc-In_4-polytopes_and_3D_honeycomb_tessellations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Degrees_of_rectification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Degrees_of_rectification"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Degrees of rectification</span> </div> </a> <button aria-controls="toc-Degrees_of_rectification-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Degrees of rectification subsection</span> </button> <ul id="toc-Degrees_of_rectification-sublist" class="vector-toc-list"> <li id="toc-Notations_and_facets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notations_and_facets"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Notations and facets</span> </div> </a> <ul id="toc-Notations_and_facets-sublist" class="vector-toc-list"> <li id="toc-Regular_polygons" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Regular_polygons"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.1</span> <span>Regular polygons</span> </div> </a> <ul id="toc-Regular_polygons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regular_polyhedra_and_tilings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Regular_polyhedra_and_tilings"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.2</span> <span>Regular polyhedra and tilings</span> </div> </a> <ul id="toc-Regular_polyhedra_and_tilings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regular_Uniform_4-polytopes_and_honeycombs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Regular_Uniform_4-polytopes_and_honeycombs"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.3</span> <span>Regular Uniform 4-polytopes and honeycombs</span> </div> </a> <ul id="toc-Regular_Uniform_4-polytopes_and_honeycombs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regular_5-polytopes_and_4-space_honeycombs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Regular_5-polytopes_and_4-space_honeycombs"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.4</span> <span>Regular 5-polytopes and 4-space honeycombs</span> </div> </a> <ul id="toc-Regular_5-polytopes_and_4-space_honeycombs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">This article is about an operation on polyhedra. For rectification of curves, see <a href="/wiki/Arc_length" title="Arc length">arc length</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Operation in Euclidean geometry</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cuboctahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Cuboctahedron.png/220px-Cuboctahedron.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Cuboctahedron.png/330px-Cuboctahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Cuboctahedron.png/440px-Cuboctahedron.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>A rectified cube is a <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a> – edges reduced to vertices, and vertices expanded into new faces</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dual_Cube-Octahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/220px-Dual_Cube-Octahedron.svg.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/330px-Dual_Cube-Octahedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Dual_Cube-Octahedron.svg/440px-Dual_Cube-Octahedron.svg.png 2x" data-file-width="744" data-file-height="749" /></a><figcaption>A <i>birectified</i> cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rectified_cubic_honeycomb.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b9/Rectified_cubic_honeycomb.jpg" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>A <a href="/wiki/Rectified_cubic_honeycomb" class="mw-redirect" title="Rectified cubic honeycomb">rectified cubic honeycomb</a> – edges reduced to vertices, and vertices expanded into new cells.</figcaption></figure> <p>In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, <b>rectification</b>, also known as <b>critical truncation</b> or <b>complete-truncation</b>, is the process of truncating a <a href="/wiki/Polytope" title="Polytope">polytope</a> by marking the midpoints of all its <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>, and cutting off its <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> at those points.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The resulting polytope will be bounded by <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> facets and the rectified facets of the original polytope. </p><p>A rectification operator is sometimes denoted by the letter <span class="texhtml mvar" style="font-style:italic;">r</span> with a <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a>. For example, <span class="nowrap"><span class="texhtml"><i>r</i>{4,3} </span></span> is the rectified <a href="/wiki/Cube" title="Cube">cube</a>, also called a <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a>, and also represented as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{Bmatrix}4\\3\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>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0fca92e771d171801da1ddd89b7ed800498a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.4ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}"></span>. And a rectified cuboctahedron <span class="texhtml">rr{4,3} </span> is a <a href="/wiki/Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a>, and also represented as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r{\begin{Bmatrix}4\\3\end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r{\begin{Bmatrix}4\\3\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af865d4b6453d442ba9ec7a438984d824857aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.449ex; height:6.176ex;" alt="{\displaystyle r{\begin{Bmatrix}4\\3\end{Bmatrix}}}"></span>. </p><p><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">Conway polyhedron notation</a> uses <span class="texhtml">a</span> for <b>ambo</b> as this operator. In <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> this operation creates a <a href="/wiki/Medial_graph" title="Medial graph">medial graph</a>. </p><p>The rectification of any regular <a href="/wiki/Self-dual" class="mw-redirect" title="Self-dual">self-dual</a> polyhedron or tiling will result in another regular polyhedron or tiling with a <a href="/w/index.php?title=Tiling_order&action=edit&redlink=1" class="new" title="Tiling order (page does not exist)">tiling order</a> of 4, for example the <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> <span class="texhtml">{3,3} </span> becoming an <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> <span class="texhtml">{3,4}.</span> As a special case, a <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a> <span class="texhtml">{4,4} </span> will turn into another square tiling <span class="texhtml">{4,4} </span> under a rectification operation. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example_of_rectification_as_a_final_truncation_to_an_edge">Example of rectification as a final truncation to an edge</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=1" title="Edit section: Example of rectification as a final truncation to an edge"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rectification is the final point of a truncation process. For example, on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form: </p><p><span typeof="mw:File"><a href="/wiki/File:Cube_truncation_sequence.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Cube_truncation_sequence.svg/520px-Cube_truncation_sequence.svg.png" decoding="async" width="520" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Cube_truncation_sequence.svg/780px-Cube_truncation_sequence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Cube_truncation_sequence.svg/1040px-Cube_truncation_sequence.svg.png 2x" data-file-width="641" data-file-height="200" /></a></span> </p> <div class="mw-heading mw-heading2"><h2 id="Higher_degree_rectifications">Higher degree rectifications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=2" title="Edit section: Higher degree rectifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the <a href="/wiki/Dual_polytope" class="mw-redirect" title="Dual polytope">dual polytope</a>. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on. </p> <div class="mw-heading mw-heading2"><h2 id="Example_of_birectification_as_a_final_truncation_to_a_face">Example of birectification as a final truncation to a face</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=3" title="Edit section: Example of birectification as a final truncation to a face"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This sequence shows a <i>birectified cube</i> as the final sequence from a cube to the dual where the original faces are truncated down to a single point: </p> <dl><dd><span typeof="mw:File"><a href="/wiki/File:Birectified_cube_sequence.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Birectified_cube_sequence.png/480px-Birectified_cube_sequence.png" decoding="async" width="480" height="524" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Birectified_cube_sequence.png/720px-Birectified_cube_sequence.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Birectified_cube_sequence.png/960px-Birectified_cube_sequence.png 2x" data-file-width="1110" data-file-height="1212" /></a></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="In_polygons">In polygons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=4" title="Edit section: In polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon. </p> <div class="mw-heading mw-heading2"><h2 id="In_polyhedra_and_plane_tilings">In polyhedra and plane tilings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=5" title="Edit section: In polyhedra and plane tilings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">quasiregular polyhedron</a></div> <p>Each <a href="/wiki/Platonic_solid" title="Platonic solid">platonic solid</a> and its <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual</a> have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.) </p><p>The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriately scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual: </p> <ul><li>The <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> is its own dual, and its rectification is the <i>tetratetrahedron</i>, better known as the <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>.</li> <li>The <a href="/wiki/Octahedron" title="Octahedron">octahedron</a> and the <a href="/wiki/Cube" title="Cube">cube</a> are each other's dual, and their rectification is the <a href="/wiki/Cuboctahedron" title="Cuboctahedron">cuboctahedron</a>.</li> <li>The <a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> and the <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a> are duals, and their rectification is the <a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a>.</li></ul> <p>Examples </p> <table class="wikitable"> <tbody><tr> <th>Family </th> <th>Parent </th> <th>Rectification </th> <th>Dual </th></tr> <tr> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />[p,q] </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th></tr> <tr align="center"> <th>[3,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-33-t0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/75px-Uniform_polyhedron-33-t0.png" decoding="async" width="75" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/113px-Uniform_polyhedron-33-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/150px-Uniform_polyhedron-33-t0.png 2x" data-file-width="632" data-file-height="681" /></a></span><br /><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-33-t1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Uniform_polyhedron-33-t1.svg/75px-Uniform_polyhedron-33-t1.svg.png" decoding="async" width="75" height="74" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Uniform_polyhedron-33-t1.svg/113px-Uniform_polyhedron-33-t1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Uniform_polyhedron-33-t1.svg/150px-Uniform_polyhedron-33-t1.svg.png 2x" data-file-width="658" data-file-height="646" /></a></span><br /><a href="/wiki/Tetratetrahedron" class="mw-redirect" title="Tetratetrahedron">Octahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-33-t2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Uniform_polyhedron-33-t2.png/75px-Uniform_polyhedron-33-t2.png" decoding="async" width="75" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Uniform_polyhedron-33-t2.png/113px-Uniform_polyhedron-33-t2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Uniform_polyhedron-33-t2.png/150px-Uniform_polyhedron-33-t2.png 2x" data-file-width="661" data-file-height="594" /></a></span><br /><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a> </td></tr> <tr align="center"> <th>[4,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-43-t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/75px-Uniform_polyhedron-43-t0.svg.png" decoding="async" width="75" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/113px-Uniform_polyhedron-43-t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/150px-Uniform_polyhedron-43-t0.svg.png 2x" data-file-width="731" data-file-height="709" /></a></span><br /><a href="/wiki/Cube" title="Cube">Cube</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-43-t1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/75px-Uniform_polyhedron-43-t1.svg.png" decoding="async" width="75" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/113px-Uniform_polyhedron-43-t1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/150px-Uniform_polyhedron-43-t1.svg.png 2x" data-file-width="1081" data-file-height="907" /></a></span><br /><a href="/wiki/Cuboctahedron" title="Cuboctahedron">Cuboctahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-43-t2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/75px-Uniform_polyhedron-43-t2.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/113px-Uniform_polyhedron-43-t2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/150px-Uniform_polyhedron-43-t2.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a> </td></tr> <tr align="center"> <th>[5,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-53-t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Uniform_polyhedron-53-t0.svg/75px-Uniform_polyhedron-53-t0.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Uniform_polyhedron-53-t0.svg/113px-Uniform_polyhedron-53-t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Uniform_polyhedron-53-t0.svg/150px-Uniform_polyhedron-53-t0.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-53-t1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Uniform_polyhedron-53-t1.svg/75px-Uniform_polyhedron-53-t1.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Uniform_polyhedron-53-t1.svg/113px-Uniform_polyhedron-53-t1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Uniform_polyhedron-53-t1.svg/150px-Uniform_polyhedron-53-t1.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_polyhedron-53-t2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Uniform_polyhedron-53-t2.svg/75px-Uniform_polyhedron-53-t2.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Uniform_polyhedron-53-t2.svg/113px-Uniform_polyhedron-53-t2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Uniform_polyhedron-53-t2.svg/150px-Uniform_polyhedron-53-t2.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a> </td></tr> <tr align="center"> <th>[6,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_63-t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Uniform_tiling_63-t0.svg/75px-Uniform_tiling_63-t0.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Uniform_tiling_63-t0.svg/113px-Uniform_tiling_63-t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Uniform_tiling_63-t0.svg/150px-Uniform_tiling_63-t0.svg.png 2x" data-file-width="640" data-file-height="640" /></a></span><br /><a href="/wiki/Hexagonal_tiling" title="Hexagonal tiling">Hexagonal tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_63-t1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Uniform_tiling_63-t1.svg/75px-Uniform_tiling_63-t1.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Uniform_tiling_63-t1.svg/113px-Uniform_tiling_63-t1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Uniform_tiling_63-t1.svg/150px-Uniform_tiling_63-t1.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span><br /><a href="/wiki/Trihexagonal_tiling" title="Trihexagonal tiling">Trihexagonal tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_63-t2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Uniform_tiling_63-t2.svg/75px-Uniform_tiling_63-t2.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Uniform_tiling_63-t2.svg/113px-Uniform_tiling_63-t2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Uniform_tiling_63-t2.svg/150px-Uniform_tiling_63-t2.svg.png 2x" data-file-width="640" data-file-height="640" /></a></span><br /><a href="/wiki/Triangular_tiling" title="Triangular tiling">Triangular tiling</a> </td></tr> <tr align="center"> <th>[7,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Heptagonal_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Heptagonal_tiling.svg/75px-Heptagonal_tiling.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Heptagonal_tiling.svg/113px-Heptagonal_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Heptagonal_tiling.svg/150px-Heptagonal_tiling.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Order-3_heptagonal_tiling" class="mw-redirect" title="Order-3 heptagonal tiling">Order-3 heptagonal tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Triheptagonal_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Triheptagonal_tiling.svg/75px-Triheptagonal_tiling.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Triheptagonal_tiling.svg/113px-Triheptagonal_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Triheptagonal_tiling.svg/150px-Triheptagonal_tiling.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Triheptagonal_tiling" title="Triheptagonal tiling">Triheptagonal tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Order-7_triangular_tiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Order-7_triangular_tiling.svg/75px-Order-7_triangular_tiling.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Order-7_triangular_tiling.svg/113px-Order-7_triangular_tiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Order-7_triangular_tiling.svg/150px-Order-7_triangular_tiling.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Order-7_triangular_tiling" title="Order-7 triangular tiling">Order-7 triangular tiling</a> </td></tr> <tr align="center"> <th>[4,4] </th> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_44-t0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Uniform_tiling_44-t0.svg/75px-Uniform_tiling_44-t0.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Uniform_tiling_44-t0.svg/113px-Uniform_tiling_44-t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Uniform_tiling_44-t0.svg/150px-Uniform_tiling_44-t0.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span><br /><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_44-t1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Uniform_tiling_44-t1.svg/75px-Uniform_tiling_44-t1.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Uniform_tiling_44-t1.svg/113px-Uniform_tiling_44-t1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Uniform_tiling_44-t1.svg/150px-Uniform_tiling_44-t1.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span><br /><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniform_tiling_44-t2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Uniform_tiling_44-t2.svg/75px-Uniform_tiling_44-t2.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Uniform_tiling_44-t2.svg/113px-Uniform_tiling_44-t2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Uniform_tiling_44-t2.svg/150px-Uniform_tiling_44-t2.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span><br /><a href="/wiki/Square_tiling" title="Square tiling">Square tiling</a> </td></tr> <tr align="center"> <th>[5,4] </th> <td><span typeof="mw:File"><a href="/wiki/File:H2-5-4-dual.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/H2-5-4-dual.svg/75px-H2-5-4-dual.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/H2-5-4-dual.svg/113px-H2-5-4-dual.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/H2-5-4-dual.svg/150px-H2-5-4-dual.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Order-4_pentagonal_tiling" title="Order-4 pentagonal tiling">Order-4 pentagonal tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2-5-4-rectified.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/H2-5-4-rectified.svg/75px-H2-5-4-rectified.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/H2-5-4-rectified.svg/113px-H2-5-4-rectified.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/H2-5-4-rectified.svg/150px-H2-5-4-rectified.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Tetrapentagonal_tiling" title="Tetrapentagonal tiling">Tetrapentagonal tiling</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H2-5-4-primal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-primal.svg/75px-H2-5-4-primal.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-primal.svg/113px-H2-5-4-primal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/H2-5-4-primal.svg/150px-H2-5-4-primal.svg.png 2x" data-file-width="2000" data-file-height="2000" /></a></span><br /><a href="/wiki/Order-5_square_tiling" title="Order-5 square tiling">Order-5 square tiling</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="In_nonregular_polyhedra">In nonregular polyhedra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=6" title="Edit section: In nonregular polyhedra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a <a href="/wiki/Polyhedral_graph" title="Polyhedral graph">polyhedral graph</a> as its <a href="/wiki/N-skeleton" title="N-skeleton">1-skeleton</a>, and from that graph one may form the <a href="/wiki/Medial_graph" title="Medial graph">medial graph</a> by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by <a href="/wiki/Steinitz%27s_theorem" title="Steinitz's theorem">Steinitz's theorem</a> it can be represented as a polyhedron. </p><p>The <a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">Conway polyhedron notation</a> equivalent to rectification is <b>ambo</b>, represented by <b>a</b>. Applying twice <b>aa</b>, (rectifying a rectification) is Conway's <a href="/wiki/Expansion_(geometry)" title="Expansion (geometry)"><b>expand</b></a> operation, <b>e</b>, which is the same as Johnson's <a href="/wiki/Cantellation_(geometry)" title="Cantellation (geometry)">cantellation</a> operation, t<sub>0,2</sub> generated from regular polyhedral and tilings. </p> <div class="mw-heading mw-heading2"><h2 id="In_4-polytopes_and_3D_honeycomb_tessellations">In 4-polytopes and 3D honeycomb tessellations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=7" title="Edit section: In 4-polytopes and 3D honeycomb tessellations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each <a href="/wiki/Convex_regular_4-polytope" class="mw-redirect" title="Convex regular 4-polytope">Convex regular 4-polytope</a> has a rectified form as a <a href="/wiki/Uniform_4-polytope" title="Uniform 4-polytope">uniform 4-polytope</a>. </p><p>A regular 4-polytope {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex. </p><p>A rectified {p,q,r} is not the same as a rectified {r,q,p}, however. A further truncation, called <a href="/wiki/Bitruncation_(geometry)" class="mw-redirect" title="Bitruncation (geometry)">bitruncation</a>, is symmetric between a 4-polytope and its dual. See <a href="/wiki/Uniform_4-polytope#Geometric_derivations" title="Uniform 4-polytope">Uniform 4-polytope#Geometric derivations</a>. </p><p>Examples </p> <table class="wikitable"> <tbody><tr> <th>Family </th> <th>Parent </th> <th>Rectification </th> <th>Birectification<br />(Dual rectification) </th> <th>Trirectification<br />(Dual) </th></tr> <tr> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />[<i>p</i>,<i>q</i>,<i>r</i>] </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />{<i>p</i>,<i>q</i>,<i>r</i>} </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />r{<i>p</i>,<i>q</i>,<i>r</i>} </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br />2r{<i>p</i>,<i>q</i>,<i>r</i>} </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span><br />3r{<i>p</i>,<i>q</i>,<i>r</i>} </th></tr> <tr align="center"> <th>[3,3,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_5-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/120px-Schlegel_wireframe_5-cell.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/180px-Schlegel_wireframe_5-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/240px-Schlegel_wireframe_5-cell.png 2x" data-file-width="817" data-file-height="785" /></a></span><br /><a href="/wiki/5-cell" title="5-cell">5-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_half-solid_rectified_5-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Schlegel_half-solid_rectified_5-cell.png/120px-Schlegel_half-solid_rectified_5-cell.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Schlegel_half-solid_rectified_5-cell.png/180px-Schlegel_half-solid_rectified_5-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Schlegel_half-solid_rectified_5-cell.png/240px-Schlegel_half-solid_rectified_5-cell.png 2x" data-file-width="1001" data-file-height="999" /></a></span><br /><a href="/wiki/Rectified_5-cell" title="Rectified 5-cell">rectified 5-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_half-solid_rectified_5-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Schlegel_half-solid_rectified_5-cell.png/120px-Schlegel_half-solid_rectified_5-cell.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Schlegel_half-solid_rectified_5-cell.png/180px-Schlegel_half-solid_rectified_5-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Schlegel_half-solid_rectified_5-cell.png/240px-Schlegel_half-solid_rectified_5-cell.png 2x" data-file-width="1001" data-file-height="999" /></a></span><br /><a href="/wiki/Rectified_5-cell" title="Rectified 5-cell">rectified 5-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_5-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/120px-Schlegel_wireframe_5-cell.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/180px-Schlegel_wireframe_5-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/240px-Schlegel_wireframe_5-cell.png 2x" data-file-width="817" data-file-height="785" /></a></span><br /><a href="/wiki/5-cell" title="5-cell">5-cell</a> </td></tr> <tr align="center"> <th>[4,3,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_8-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/150px-Schlegel_wireframe_8-cell.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/225px-Schlegel_wireframe_8-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/300px-Schlegel_wireframe_8-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Tesseract" title="Tesseract">tesseract</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_half-solid_rectified_8-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_half-solid_rectified_8-cell.png/150px-Schlegel_half-solid_rectified_8-cell.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_half-solid_rectified_8-cell.png/225px-Schlegel_half-solid_rectified_8-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_half-solid_rectified_8-cell.png/300px-Schlegel_half-solid_rectified_8-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Rectified_tesseract" title="Rectified tesseract">rectified tesseract</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_half-solid_rectified_16-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Schlegel_half-solid_rectified_16-cell.png/150px-Schlegel_half-solid_rectified_16-cell.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Schlegel_half-solid_rectified_16-cell.png/225px-Schlegel_half-solid_rectified_16-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Schlegel_half-solid_rectified_16-cell.png/300px-Schlegel_half-solid_rectified_16-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br />Rectified 16-cell<br />(<a href="/wiki/24-cell" title="24-cell">24-cell</a>) </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_16-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/150px-Schlegel_wireframe_16-cell.png" decoding="async" width="150" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/225px-Schlegel_wireframe_16-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Schlegel_wireframe_16-cell.png/300px-Schlegel_wireframe_16-cell.png 2x" data-file-width="768" data-file-height="786" /></a></span><br /><a href="/wiki/16-cell" title="16-cell">16-cell</a> </td></tr> <tr align="center"> <th>[3,4,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_24-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/150px-Schlegel_wireframe_24-cell.png" decoding="async" width="150" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/225px-Schlegel_wireframe_24-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/300px-Schlegel_wireframe_24-cell.png 2x" data-file-width="862" data-file-height="816" /></a></span><br /><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_half-solid_cantellated_16-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Schlegel_half-solid_cantellated_16-cell.png/150px-Schlegel_half-solid_cantellated_16-cell.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Schlegel_half-solid_cantellated_16-cell.png/225px-Schlegel_half-solid_cantellated_16-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Schlegel_half-solid_cantellated_16-cell.png/300px-Schlegel_half-solid_cantellated_16-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Rectified_24-cell" title="Rectified 24-cell">rectified 24-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_half-solid_cantellated_16-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Schlegel_half-solid_cantellated_16-cell.png/150px-Schlegel_half-solid_cantellated_16-cell.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Schlegel_half-solid_cantellated_16-cell.png/225px-Schlegel_half-solid_cantellated_16-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Schlegel_half-solid_cantellated_16-cell.png/300px-Schlegel_half-solid_cantellated_16-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Rectified_24-cell" title="Rectified 24-cell">rectified 24-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_24-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/150px-Schlegel_wireframe_24-cell.png" decoding="async" width="150" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/225px-Schlegel_wireframe_24-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Schlegel_wireframe_24-cell.png/300px-Schlegel_wireframe_24-cell.png 2x" data-file-width="862" data-file-height="816" /></a></span><br /><a href="/wiki/24-cell" title="24-cell">24-cell</a> </td></tr> <tr align="center"> <th>[5,3,3] </th> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_120-cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Schlegel_wireframe_120-cell.png/150px-Schlegel_wireframe_120-cell.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Schlegel_wireframe_120-cell.png/225px-Schlegel_wireframe_120-cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Schlegel_wireframe_120-cell.png/300px-Schlegel_wireframe_120-cell.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/120-cell" title="120-cell">120-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rectified_120-cell_schlegel_halfsolid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Rectified_120-cell_schlegel_halfsolid.png/150px-Rectified_120-cell_schlegel_halfsolid.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Rectified_120-cell_schlegel_halfsolid.png/225px-Rectified_120-cell_schlegel_halfsolid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Rectified_120-cell_schlegel_halfsolid.png/300px-Rectified_120-cell_schlegel_halfsolid.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Rectified_120-cell" title="Rectified 120-cell">rectified 120-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rectified_600-cell_schlegel_halfsolid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Rectified_600-cell_schlegel_halfsolid.png/150px-Rectified_600-cell_schlegel_halfsolid.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Rectified_600-cell_schlegel_halfsolid.png/225px-Rectified_600-cell_schlegel_halfsolid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Rectified_600-cell_schlegel_halfsolid.png/300px-Rectified_600-cell_schlegel_halfsolid.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Rectified_600-cell" title="Rectified 600-cell">rectified 600-cell</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Schlegel_wireframe_600-cell_vertex-centered.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_wireframe_600-cell_vertex-centered.png/150px-Schlegel_wireframe_600-cell_vertex-centered.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_wireframe_600-cell_vertex-centered.png/225px-Schlegel_wireframe_600-cell_vertex-centered.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Schlegel_wireframe_600-cell_vertex-centered.png/300px-Schlegel_wireframe_600-cell_vertex-centered.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/600-cell" title="600-cell">600-cell</a> </td></tr> <tr align="center"> <th>[4,3,4] </th> <td><span typeof="mw:File"><a href="/wiki/File:Partial_cubic_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/150px-Partial_cubic_honeycomb.png" decoding="async" width="150" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/225px-Partial_cubic_honeycomb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/300px-Partial_cubic_honeycomb.png 2x" data-file-width="873" data-file-height="866" /></a></span><br /><a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">Cubic honeycomb</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rectified_cubic_honeycomb.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Rectified_cubic_honeycomb.jpg/150px-Rectified_cubic_honeycomb.jpg" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b9/Rectified_cubic_honeycomb.jpg 1.5x" data-file-width="200" data-file-height="200" /></a></span><br /><a href="/wiki/Rectified_cubic_honeycomb" class="mw-redirect" title="Rectified cubic honeycomb">Rectified cubic honeycomb</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rectified_cubic_honeycomb.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Rectified_cubic_honeycomb.jpg/150px-Rectified_cubic_honeycomb.jpg" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b9/Rectified_cubic_honeycomb.jpg 1.5x" data-file-width="200" data-file-height="200" /></a></span><br /><a href="/wiki/Rectified_cubic_honeycomb" class="mw-redirect" title="Rectified cubic honeycomb">Rectified cubic honeycomb</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Partial_cubic_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/150px-Partial_cubic_honeycomb.png" decoding="async" width="150" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/225px-Partial_cubic_honeycomb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/300px-Partial_cubic_honeycomb.png 2x" data-file-width="873" data-file-height="866" /></a></span><br /><a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">Cubic honeycomb</a> </td></tr> <tr align="center"> <th>[5,3,4] </th> <td><span typeof="mw:File"><a href="/wiki/File:Hyperbolic_orthogonal_dodecahedral_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Hyperbolic_orthogonal_dodecahedral_honeycomb.png/150px-Hyperbolic_orthogonal_dodecahedral_honeycomb.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Hyperbolic_orthogonal_dodecahedral_honeycomb.png/225px-Hyperbolic_orthogonal_dodecahedral_honeycomb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Hyperbolic_orthogonal_dodecahedral_honeycomb.png/300px-Hyperbolic_orthogonal_dodecahedral_honeycomb.png 2x" data-file-width="902" data-file-height="902" /></a></span><br /><a href="/wiki/Order-4_dodecahedral_honeycomb" title="Order-4 dodecahedral honeycomb">Order-4 dodecahedral</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rectified_order_4_dodecahedral_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Rectified_order_4_dodecahedral_honeycomb.png/150px-Rectified_order_4_dodecahedral_honeycomb.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Rectified_order_4_dodecahedral_honeycomb.png/225px-Rectified_order_4_dodecahedral_honeycomb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Rectified_order_4_dodecahedral_honeycomb.png/300px-Rectified_order_4_dodecahedral_honeycomb.png 2x" data-file-width="900" data-file-height="900" /></a></span><br /><a href="/wiki/Rectified_order-4_dodecahedral_honeycomb" class="mw-redirect" title="Rectified order-4 dodecahedral honeycomb">Rectified order-4 dodecahedral</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:H3_435_CC_center_0100.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/H3_435_CC_center_0100.png/150px-H3_435_CC_center_0100.png" decoding="async" width="150" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/H3_435_CC_center_0100.png/225px-H3_435_CC_center_0100.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/H3_435_CC_center_0100.png/300px-H3_435_CC_center_0100.png 2x" data-file-width="1024" data-file-height="768" /></a></span><br /><a href="/wiki/Rectified_order-5_cubic_honeycomb" class="mw-redirect" title="Rectified order-5 cubic honeycomb">Rectified order-5 cubic</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hyperb_gcubic_hc.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Hyperb_gcubic_hc.png/150px-Hyperb_gcubic_hc.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Hyperb_gcubic_hc.png/225px-Hyperb_gcubic_hc.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Hyperb_gcubic_hc.png/300px-Hyperb_gcubic_hc.png 2x" data-file-width="800" data-file-height="800" /></a></span><br /><a href="/wiki/Order-5_cubic_honeycomb" title="Order-5 cubic honeycomb">Order-5 cubic</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Degrees_of_rectification">Degrees of rectification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=8" title="Edit section: Degrees of rectification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A first rectification truncates edges down to points. If a polytope is <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a>, this form is represented by an extended <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> notation <b>t</b><sub>1</sub>{p,q,...} or <b>r</b>{p,q,...}. </p><p>A second rectification, or <b>birectification</b>, truncates <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> down to points. If regular it has notation <b>t</b><sub>2</sub>{p,q,...} or 2<b>r</b>{p,q,...}. For <a href="/wiki/Polyhedron" title="Polyhedron">polyhedra</a>, a birectification creates a <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a>. </p><p>Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates <i>n-faces</i> to points. </p><p>If an n-polytope is (n-1)-rectified, its <a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">facets</a> are reduced to points and the polytope becomes its <a href="/wiki/Dual_polytope" class="mw-redirect" title="Dual polytope">dual</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Notations_and_facets">Notations and facets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=9" title="Edit section: Notations and facets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of <a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">facets</a> for each. </p> <div class="mw-heading mw-heading4"><h4 id="Regular_polygons">Regular <a href="/wiki/Polygon" title="Polygon">polygons</a></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=10" title="Edit section: Regular polygons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">Facets</a> are edges, represented as {}. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">name<br />{p} </th> <th rowspan="2"><a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter diagram</a> </th> <th rowspan="2">t-notation<br /><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <th colspan="3">Vertical <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th></tr> <tr> <th>Name </th> <th>Facet-1 </th> <th>Facet-2 </th></tr> <tr align="center"> <td>Parent </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>0</sub>{p} </td> <td>{p} </td> <td>{} </td> <td> </td></tr> <tr align="center"> <td>Rectified </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </td> <td>t<sub>1</sub>{p} </td> <td>{p} </td> <td> </td> <td>{} </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Regular_polyhedra_and_tilings">Regular <a href="/wiki/Uniform_polyhedron" title="Uniform polyhedron">polyhedra</a> and <a href="/wiki/List_of_uniform_tilings" class="mw-redirect" title="List of uniform tilings">tilings</a></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=11" title="Edit section: Regular polyhedra and tilings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">Facets</a> are regular polygons. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">name<br />{p,q} </th> <th rowspan="2"><a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter diagram</a> </th> <th rowspan="2">t-notation<br /><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <th colspan="3">Vertical <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th></tr> <tr> <th>Name </th> <th>Facet-1 </th> <th>Facet-2 </th></tr> <tr align="center"> <td>Parent </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/3/36/CDel_split1-pq.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/b2/CDel_nodes_10lu.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </td> <td>t<sub>0</sub>{p,q} </td> <td>{p,q} </td> <td>{p} </td> <td> </td></tr> <tr align="center"> <td>Rectified </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/3/36/CDel_split1-pq.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>1</sub>{p,q} </td> <td>r{p,q} = <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}p\\q\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> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b44b24b6cc5f609271f96f83314e37a393ee23b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.407ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}"></span> </td> <td>{p} </td> <td>{q} </td></tr> <tr align="center"> <td>Birectified </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/3/36/CDel_split1-pq.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5a/CDel_nodes_01ld.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </td> <td>t<sub>2</sub>{p,q} </td> <td>{q,p} </td> <td> </td> <td>{q} </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Regular_Uniform_4-polytopes_and_honeycombs">Regular <a href="/wiki/Uniform_4-polytope" title="Uniform 4-polytope">Uniform 4-polytopes</a> and <a href="/wiki/Honeycomb" title="Honeycomb">honeycombs</a></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=12" title="Edit section: Regular Uniform 4-polytopes and honeycombs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">Facets</a> are regular or rectified polyhedra. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">name<br />{p,q,r} </th> <th rowspan="2"><a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter diagram</a> </th> <th rowspan="2">t-notation<br /><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <th colspan="3">Extended <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th></tr> <tr> <th>Name </th> <th>Facet-1 </th> <th>Facet-2 </th></tr> <tr align="center"> <td>Parent </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>0</sub>{p,q,r} </td> <td>{p,q,r} </td> <td>{p,q} </td> <td> </td></tr> <tr align="center"> <td>Rectified </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>1</sub>{p,q,r} </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}p\ \ \\q,r\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> <mi>p</mi> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> <mo>,</mo> <mi>r</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}p\ \ \\q,r\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e699b556125a0cd929377620462751b16136f7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.39ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}p\ \ \\q,r\end{Bmatrix}}}"></span> = r{p,q,r} </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}p\\q\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> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b44b24b6cc5f609271f96f83314e37a393ee23b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.407ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}"></span> = r{p,q} </td> <td>{q,r} </td></tr> <tr align="center"> <td>Birectified<br />(Dual rectified) </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>2</sub>{p,q,r} </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}q,p\\r\ \ \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> <mi>q</mi> <mo>,</mo> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}q,p\\r\ \ \end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbfd0ccf626e3d697f070e78e6eb962000a434e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.511ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}q,p\\r\ \ \end{Bmatrix}}}"></span> = r{r,q,p} </td> <td>{q,r} </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}q\\r\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> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}q\\r\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63aa0ae57596f64b8ecb6253871b0f0d3fcd33e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.307ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}q\\r\end{Bmatrix}}}"></span> = r{q,r} </td></tr> <tr align="center"> <td>Trirectified<br />(Dual) </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </td> <td>t<sub>3</sub>{p,q,r} </td> <td>{r,q,p} </td> <td> </td> <td>{r,q} </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Regular_5-polytopes_and_4-space_honeycombs">Regular <a href="/wiki/5-polytope" title="5-polytope">5-polytopes</a> and 4-space <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">honeycombs</a></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=13" title="Edit section: Regular 5-polytopes and 4-space honeycombs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">Facets</a> are regular or rectified 4-polytopes. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">name<br />{p,q,r,s} </th> <th rowspan="2"><a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter diagram</a> </th> <th rowspan="2">t-notation<br /><a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th> <th colspan="3">Extended <a href="/wiki/Schl%C3%A4fli_symbol" title="Schläfli symbol">Schläfli symbol</a> </th></tr> <tr> <th>Name </th> <th>Facet-1 </th> <th>Facet-2 </th></tr> <tr align="center"> <td>Parent </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>0</sub>{p,q,r,s} </td> <td>{p,q,r,s} </td> <td>{p,q,r} </td> <td> </td></tr> <tr align="center"> <td>Rectified </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>1</sub>{p,q,r,s} </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}p\ \ \ \ \ \\q,r,s\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> <mi>p</mi> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}p\ \ \ \ \ \\q,r,s\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de00d0b18116776c1808a9b5b231cf8f84aa4502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.514ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}p\ \ \ \ \ \\q,r,s\end{Bmatrix}}}"></span> = r{p,q,r,s} </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}p\ \ \\q,r\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> <mi>p</mi> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> <mo>,</mo> <mi>r</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}p\ \ \\q,r\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e699b556125a0cd929377620462751b16136f7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.39ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}p\ \ \\q,r\end{Bmatrix}}}"></span> = r{p,q,r} </td> <td>{q,r,s} </td></tr> <tr align="center"> <td>Birectified<br />(Birectified dual) </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>2</sub>{p,q,r,s} </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}q,p\\r,s\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> <mi>q</mi> <mo>,</mo> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}q,p\\r,s\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b28f5d9962d0d07a3a437e4a2b1f349db99bd758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.511ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}q,p\\r,s\end{Bmatrix}}}"></span> = 2r{p,q,r,s} </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}q,p\\r\ \ \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> <mi>q</mi> <mo>,</mo> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}q,p\\r\ \ \end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbfd0ccf626e3d697f070e78e6eb962000a434e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.511ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}q,p\\r\ \ \end{Bmatrix}}}"></span> = r{r,q,p} </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}q\ \ \\r,s\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> <mi>q</mi> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}q\ \ \\r,s\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e34b7153667f242d8509cfc56af8ae44c5ec216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.411ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}q\ \ \\r,s\end{Bmatrix}}}"></span> = r{q,r,s} </td></tr> <tr align="center"> <td>Trirectified<br />(Rectified dual) </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </td> <td>t<sub>3</sub>{p,q,r,s} </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}r,q,p\\s\ \ \ \ \ \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> <mi>r</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}r,q,p\\s\ \ \ \ \ \end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eae106756675867195a267537c360a272275ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.593ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}r,q,p\\s\ \ \ \ \ \end{Bmatrix}}}"></span> = r{s,r,q,p} </td> <td>{r,q,p} </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}r,q\\s\ \ \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> <mi>r</mi> <mo>,</mo> <mi>q</mi> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> <mtext> </mtext> <mtext> </mtext> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}r,q\\s\ \ \end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7cab4229c57aee4f32f9fc5543a3554867faaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.39ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}r,q\\s\ \ \end{Bmatrix}}}"></span> = r{s,r,q} </td></tr> <tr align="center"> <td>Quadrirectified<br />(Dual) </td> <td><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </td> <td>t<sub>4</sub>{p,q,r,s} </td> <td>{s,r,q,p} </td> <td> </td> <td>{s,r,q} </td></tr></tbody></table> <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=Rectification_(geometry)&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dual_polytope" class="mw-redirect" title="Dual polytope">Dual polytope</a></li> <li><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">Quasiregular polyhedron</a></li> <li><a href="/wiki/List_of_regular_polytopes" title="List of regular polytopes">List of regular polytopes</a></li> <li><a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">Truncation (geometry)</a></li> <li><a href="/wiki/Conway_polyhedron_notation" title="Conway polyhedron notation">Conway polyhedron notation</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=Rectification_(geometry)&action=edit&section=15" 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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Rectification"><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="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Rectification.html">"Rectification"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Rectification&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRectification.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARectification+%28geometry%29" class="Z3988"></span></span></span> </li> </ol></div></div> <ul><li><a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter, H.S.M.</a> <i><a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)">Regular Polytopes</a></i>, (3rd edition, 1973), Dover edition, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61480-8" title="Special:BookSources/0-486-61480-8">0-486-61480-8</a> (pp. 145–154 Chapter 8: Truncation)</li> <li><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">Norman Johnson</a> <i>Uniform Polytopes</i>, Manuscript (1991) <ul><li><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">N.W. Johnson</a>: <i>The Theory of Uniform Polytopes and Honeycombs</i>, Ph.D. Dissertation, University of Toronto, 1966</li></ul></li> <li><a href="/wiki/John_Horton_Conway" title="John Horton Conway">John H. Conway</a>, Heidi Burgiel, Chaim Goodman-Strauss, <i>The Symmetries of Things</i> 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-220-5" title="Special:BookSources/978-1-56881-220-5">978-1-56881-220-5</a> (Chapter 26)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rectification_(geometry)&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOlshevsky" class="citation web cs1">Olshevsky, George. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Rectification">"Rectification"</a>. <i>Glossary for Hyperspace</i>. Archived from <a rel="nofollow" class="external text" href="http://members.aol.com/Polycell/glossary.html#Rectification">the original</a> on 4 February 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Glossary+for+Hyperspace&rft.atitle=Rectification&rft.aulast=Olshevsky&rft.aufirst=George&rft_id=http%3A%2F%2Fmembers.aol.com%2FPolycell%2Fglossary.html%23Rectification&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARectification+%28geometry%29" class="Z3988"></span></li></ul> <table class="wikitable" style="text-align:center;"> <caption>Polyhedron operators <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 ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini" style="float:right;"><ul class="navbar-brackets"><li class="nv-view"><a href="/wiki/Template:Polyhedron_operators" title="Template:Polyhedron operators"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polyhedron_operators" title="Template talk:Polyhedron operators"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polyhedron_operators" title="Special:EditPage/Template:Polyhedron operators"><abbr title="Edit this template">e</abbr></a></li></ul></div> </caption> <tbody><tr> <th>Seed </th> <th><small><a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">Truncation</a></small> </th> <th><small><a class="mw-selflink selflink">Rectification</a></small> </th> <th><small><a href="/wiki/Bitruncation" title="Bitruncation">Bitruncation</a></small> </th> <th><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">Dual</a> </th> <th><small><a href="/wiki/Expansion_(geometry)" title="Expansion (geometry)">Expansion</a></small> </th> <th><small><a href="/wiki/Omnitruncation" title="Omnitruncation">Omnitruncation</a></small> </th> <th colspan="3"><a href="/wiki/Alternation_(geometry)" title="Alternation (geometry)">Alternations</a> </th></tr> <tr> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img 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class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th> <th><span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/Regular_polyhedron" title="Regular polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/50px-Uniform_polyhedron-43-t0.svg.png" decoding="async" width="50" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/75px-Uniform_polyhedron-43-t0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/100px-Uniform_polyhedron-43-t0.svg.png 2x" data-file-width="731" data-file-height="709" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Truncated_polyhedron" title="Truncated polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Uniform_polyhedron-43-t01.svg/50px-Uniform_polyhedron-43-t01.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Uniform_polyhedron-43-t01.svg/75px-Uniform_polyhedron-43-t01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Uniform_polyhedron-43-t01.svg/100px-Uniform_polyhedron-43-t01.svg.png 2x" data-file-width="799" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/50px-Uniform_polyhedron-43-t1.svg.png" decoding="async" width="50" height="42" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/75px-Uniform_polyhedron-43-t1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/100px-Uniform_polyhedron-43-t1.svg.png 2x" data-file-width="1081" data-file-height="907" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Bitruncated_polyhedron" title="Bitruncated polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Uniform_polyhedron-43-t12.svg/50px-Uniform_polyhedron-43-t12.svg.png" decoding="async" width="50" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Uniform_polyhedron-43-t12.svg/75px-Uniform_polyhedron-43-t12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Uniform_polyhedron-43-t12.svg/100px-Uniform_polyhedron-43-t12.svg.png 2x" data-file-width="750" data-file-height="728" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/50px-Uniform_polyhedron-43-t2.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/75px-Uniform_polyhedron-43-t2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/100px-Uniform_polyhedron-43-t2.svg.png 2x" data-file-width="800" data-file-height="800" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Cantellated_polyhedron" title="Cantellated polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Uniform_polyhedron-43-t02.png/50px-Uniform_polyhedron-43-t02.png" decoding="async" width="50" height="52" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Uniform_polyhedron-43-t02.png/75px-Uniform_polyhedron-43-t02.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Uniform_polyhedron-43-t02.png/100px-Uniform_polyhedron-43-t02.png 2x" data-file-width="738" data-file-height="764" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Omnitruncated_polyhedron" title="Omnitruncated polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Uniform_polyhedron-43-t012.png/50px-Uniform_polyhedron-43-t012.png" decoding="async" width="50" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Uniform_polyhedron-43-t012.png/75px-Uniform_polyhedron-43-t012.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Uniform_polyhedron-43-t012.png/100px-Uniform_polyhedron-43-t012.png 2x" data-file-width="782" data-file-height="763" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Alternation_(geometry)" title="Alternation (geometry)"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/50px-Uniform_polyhedron-33-t0.png" decoding="async" width="50" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/75px-Uniform_polyhedron-33-t0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/100px-Uniform_polyhedron-33-t0.png 2x" data-file-width="632" data-file-height="681" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Snub_polyhedron" title="Snub polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/50px-Uniform_polyhedron-43-h01.svg.png" decoding="async" width="50" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/75px-Uniform_polyhedron-43-h01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/100px-Uniform_polyhedron-43-h01.svg.png 2x" data-file-width="240" data-file-height="217" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/Snub_polyhedron" title="Snub polyhedron"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Uniform_polyhedron-43-s012.png/50px-Uniform_polyhedron-43-s012.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Uniform_polyhedron-43-s012.png/75px-Uniform_polyhedron-43-s012.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Uniform_polyhedron-43-s012.png/100px-Uniform_polyhedron-43-s012.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td></tr> <tr> <td><span class="texhtml"><a href="/wiki/Regular_polyhedron" title="Regular polyhedron">t<sub>0</sub>{<i>p</i>,<i>q</i>}</a><br />{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Truncated_polyhedron" class="mw-redirect" title="Truncated polyhedron">t<sub>01</sub>{<i>p</i>,<i>q</i>}</a><br />t{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Quasiregular_polyhedron" title="Quasiregular polyhedron">t<sub>1</sub>{<i>p</i>,<i>q</i>}</a><br />r{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Bitruncated_polyhedron" class="mw-redirect" title="Bitruncated polyhedron">t<sub>12</sub>{<i>p</i>,<i>q</i>}</a><br />2t{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Dual_polyhedron" title="Dual polyhedron">t<sub>2</sub>{<i>p</i>,<i>q</i>}</a><br />2r{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Cantellated_polyhedron" class="mw-redirect" title="Cantellated polyhedron">t<sub>02</sub>{<i>p</i>,<i>q</i>}</a><br />rr{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Omnitruncated_polyhedron" title="Omnitruncated polyhedron">t<sub>012</sub>{<i>p</i>,<i>q</i>}</a><br />tr{<i>p</i>,<i>q</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Alternation_(geometry)" title="Alternation (geometry)">ht<sub>0</sub>{<i>p</i>,<i>q</i>}</a><br />h{<i>q</i>,<i>p</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Snub_polyhedron" title="Snub polyhedron">ht<sub>12</sub>{<i>p</i>,<i>q</i>}</a><br />s{<i>q</i>,<i>p</i>} </span> </td> <td><span class="texhtml"><a href="/wiki/Snub_polyhedron" title="Snub polyhedron">ht<sub>012</sub>{<i>p</i>,<i>q</i>}</a><br />sr{<i>p</i>,<i>q</i>} </span> </td></tr></tbody></table>    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