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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Expansion (geometry)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://it.wikipedia.org/wiki/Espansione_(geometria)" title="Espansione (geometria) – Italian" lang="it" hreflang="it" data-title="Espansione (geometria)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Expans%C3%A3o_de_s%C3%B3lido" title="Expansão de sólido – Portuguese" lang="pt" hreflang="pt" data-title="Expansão de sólido" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Expandare_(geometrie)" title="Expandare (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Expandare (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" 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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">Not to be confused with <a href="/wiki/Enlargement_(geometry)" class="mw-redirect" title="Enlargement (geometry)">enlargement (geometry)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Expanding_pentagon.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Expanding_pentagon.png/220px-Expanding_pentagon.png" decoding="async" width="220" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Expanding_pentagon.png/330px-Expanding_pentagon.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Expanding_pentagon.png/440px-Expanding_pentagon.png 2x" data-file-width="474" data-file-height="536" /></a><figcaption>An example of expanding <a href="/wiki/Pentagon" title="Pentagon">pentagon</a> into a <a href="/wiki/Decagon" title="Decagon">decagon</a> by moving edges away from the center and inserting new edges in the gaps. The <i>expansion</i> is <i>uniform</i> if all the edges are the same length.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:P2-A5-P3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/79/P2-A5-P3.gif" decoding="async" width="160" height="160" class="mw-file-element" data-file-width="160" data-file-height="160" /></a><figcaption>Animation showing an expanded <a href="/wiki/Cube" title="Cube">cube</a> (and <a href="/wiki/Octahedron" title="Octahedron">octahedron</a>)</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, <b>expansion</b> is a <a href="/wiki/Polytope" title="Polytope">polytope</a> operation where <a href="/wiki/Facet_(mathematics)" class="mw-redirect" title="Facet (mathematics)">facets</a> are separated and moved radially apart, and new facets are formed at separated elements (<a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>, <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size. </p><p>The expansion of a <a href="/wiki/Regular_polytope" title="Regular polytope">regular</a> <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytope</a> creates a <a href="/wiki/Uniform_polytope" title="Uniform polytope">uniform</a> convex polytope. </p><p>For polyhedra, an expanded polyhedron has all the <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> of the original polyhedron, all the faces of the <a href="/wiki/Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a>, and new square faces in place of the original edges. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Expansion_of_regular_polytopes">Expansion of regular polytopes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expansion_(geometry)&action=edit&section=1" title="Edit section: Expansion of regular polytopes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a>, this multidimensional term was defined by <a href="/wiki/Alicia_Boole_Stott" title="Alicia Boole Stott">Alicia Boole Stott</a><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> for creating new polytopes, specifically starting from <a href="/wiki/Regular_polytope" title="Regular polytope">regular polytopes</a> to construct new <a href="/wiki/Uniform_polytope" title="Uniform polytope">uniform polytopes</a>. </p><p>The <i>expansion</i> operation is symmetric with respect to a regular polytope and its <a href="/wiki/Dual_polyhedron#Dual_polytopes_and_tessellations" title="Dual polyhedron">dual</a>. The resulting figure contains the <a href="/wiki/Facet_(geometry)" title="Facet (geometry)">facets</a> of both the regular and its dual, along with various prismatic facets filling the gaps created between intermediate dimensional elements. </p><p>It has somewhat different meanings by <a href="/wiki/Dimension" title="Dimension">dimension</a>. In a <a href="/wiki/Wythoff_construction" title="Wythoff construction">Wythoff construction</a>, an expansion is generated by reflections from the first and last mirrors. In higher dimensions, lower dimensional expansions can be written with a subscript, so e<sub>2</sub> is the same as t<sub>0,2</sub> in any dimension. </p><p>By dimension: </p> <ul><li>A regular {p} <a href="/wiki/Polygon" title="Polygon">polygon</a> expands into a regular 2n-gon. <ul><li>The operation is identical to <a href="/wiki/Truncation_(geometry)" title="Truncation (geometry)">truncation</a> for polygons, e{p} = e<sub>1</sub>{p} = t<sub>0,1</sub>{p} = t{p} and has <a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter-Dynkin diagram</a> <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>.</li></ul></li> <li>A regular {p,q} <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> (3-polytope) expands into a polyhedron with <a href="/wiki/Vertex_figure" title="Vertex figure">vertex figure</a> <i>p.4.q.4</i>. <ul><li>This operation for polyhedra is also called <a href="/wiki/Cantellation_(geometry)" title="Cantellation (geometry)">cantellation</a>, e{p,q} = e<sub>2</sub>{p,q} = t<sub>0,2</sub>{p,q} = rr{p,q}, and has Coxeter diagram <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>. <dl><dd><span typeof="mw:File"><a href="/wiki/File:Cube_cantellation_sequence.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Cube_cantellation_sequence.svg/480px-Cube_cantellation_sequence.svg.png" decoding="async" width="480" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Cube_cantellation_sequence.svg/720px-Cube_cantellation_sequence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Cube_cantellation_sequence.svg/960px-Cube_cantellation_sequence.svg.png 2x" data-file-width="870" data-file-height="250" /></a></span></dd> <dd>For example, a rhombicuboctahedron can be called an <i>expanded cube</i>, <i>expanded octahedron</i>, as well as a <i>cantellated cube</i> or <i>cantellated octahedron</i>.</dd></dl></li></ul></li> <li>A regular {p,q,r} <a href="/wiki/4-polytope" title="4-polytope">4-polytope</a> (4-polytope) expands into a new 4-polytope with the original {p,q} cells, new cells {r,q} in place of the old vertices, p-gonal prisms in place of the old faces, and r-gonal prisms in place of the old edges. <ul><li>This operation for 4-polytopes is also called <a href="/wiki/Runcination_(geometry)" class="mw-redirect" title="Runcination (geometry)">runcination</a>, e{p,q,r} = e<sub>3</sub>{p,q,r} = t<sub>0,3</sub>{p,q,r}, and has Coxeter diagram <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/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>.</li></ul></li> <li>Similarly a regular {p,q,r,s} <a href="/wiki/5-polytope" title="5-polytope">5-polytope</a> expands into a new 5-polytope with facets {p,q,r}, {s,r,q}, {p,q}×{ } <a href="/wiki/Uniform_4-polytope#Prismatic_uniform_s" title="Uniform 4-polytope">prisms</a>, {s,r}×{ } prisms, and {p}<i>×</i>{s} <a href="/wiki/Duoprism" title="Duoprism">duoprisms</a>. <ul><li>This operation is called <a href="/wiki/Sterication" class="mw-redirect" title="Sterication">sterication</a>, e{p,q,r,s} = e<sub>4</sub>{p,q,r,s} = t<sub>0,4</sub>{p,q,r,s} = 2r2r{p,q,r,s} and has Coxeter diagram <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/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>.</li></ul></li></ul> <p>The general operator for expansion of a regular n-polytope is t<sub>0,n-1</sub>{p,q,r,...}. New regular facets are added at each vertex, and new prismatic polytopes are added at each divided edge, face, ... <a href="/wiki/Ridge_(geometry)" class="mw-redirect" title="Ridge (geometry)">ridge</a>, etc. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expansion_(geometry)&action=edit&section=2" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expansion_(geometry)&action=edit&section=3" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><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">Coxeter, <i>Regular Polytopes</i> (1973), p. 123. p.210</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Expansion_(geometry)&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Expansion"><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/Expansion.html">"Expansion"</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=Expansion&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FExpansion.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExpansion+%28geometry%29" class="Z3988"></span></span></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a>, <i><a href="/wiki/Regular_Polytopes_(book)" title="Regular Polytopes (book)">Regular Polytopes</a></i>. 3rd edition, Dover, (1973) <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>.</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></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 li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini" 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 href="/wiki/Rectification_(geometry)" title="Rectification (geometry)">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 class="mw-selflink selflink">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 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/6/6d/CDel_node_n1.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/2/20/CDel_node_n2.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/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" 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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>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Expansion_(geometry)&oldid=1206626666">https://en.wikipedia.org/w/index.php?title=Expansion_(geometry)&oldid=1206626666</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a 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