Truncated 5-simplexes

<h2 id="truncated-5-simplex">Truncated 5-simplex</h2>
<table><tbody><tr><td colspan="3">Truncated 5-simplex</td></tr><tr><td>Type</td><td colspan="2"><a href="/facts/Uniform_5-polytope/mqxsoqwY">Uniform 5-polytope</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td colspan="2">t{3,3,3,3}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a></td><td colspan="2"></td></tr><tr><td>4-faces</td><td>12</td><td>6 <a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a>6 <a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td></tr><tr><td>Cells</td><td>45</td><td>30 <a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a>15 <a href="/facts/Truncated_tetrahedron/K9ItvmUK">t{3,3}</a></td></tr><tr><td>Faces</td><td>80</td><td>60 <a href="/facts/Triangle/cRXUwsMn">{3}</a>20 <a href="/facts/Hexagon/DPzuCzWE">{6}</a></td></tr><tr><td>Edges</td><td colspan="2">75</td></tr><tr><td>Vertices</td><td colspan="2">30</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td colspan="2"><a href="/facts/Tetrahedral_pyramid/Rh6lwDcc">( )v{3,3}</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td colspan="2">A5 [3,3,3,3], order 720</td></tr><tr><td>Properties</td><td colspan="2"><a href="/facts/Convex_polytope/2pRHcRgC">convex</a></td></tr></tbody></table>
<p>The truncated 5-simplex has 30 <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a>, 75 <a href="/facts/Edge_(geometry)/3lFN6sEz">edges</a>, 80 <a href="/facts/Triangle/cRXUwsMn">triangular</a> <a href="/facts/Face_(geometry)/T2E6BTU4">faces</a>, 45 <a href="/facts/Cell_(geometry)/T2E6BTU4">cells</a> (15 <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedral</a>, and 30 <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a>), and 12 <a href="/facts/4-face/T2E6BTU4">4-faces</a> (6 <a href="/facts/5-cell/Rh6lwDcc">5-cell</a> and 6 <a href="/facts/Truncated_5-cell/oqF6jkb7">truncated 5-cells</a>).
</p>
<h3>Alternate names</h3>
<ul><li>Truncated hexateron (Acronym: tix) (Jonathan Bowers)<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul>
<h3>Coordinates</h3>
<p>The vertices of the <i>truncated 5-simplex</i> can be most simply constructed on a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> in 6-space as permutations of (0,0,0,0,1,2) <i>or</i> of (0,1,2,2,2,2). These coordinates come from facets of the <a href="/facts/Truncated_6-orthoplex/xwdfvuzF">truncated 6-orthoplex</a> and <a href="/facts/Bitruncated_6-cube/uDq8zaRk">bitruncated 6-cube</a> respectively.
</p>
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th>Ak<a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>A5</th><th>A4</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[6]</td><td>[5]</td></tr><tr><th>Ak<a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>A3</th><th>A2</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[4]</td><td>[3]</td></tr></tbody></table>
<h2 id="bitruncated-5-simplex">Bitruncated 5-simplex</h2>
<table><tbody><tr><td colspan="3">bitruncated 5-simplex</td></tr><tr><td>Type</td><td colspan="2"><a href="/facts/Uniform_5-polytope/mqxsoqwY">Uniform 5-polytope</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td colspan="2">2t{3,3,3,3}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a></td><td colspan="2"></td></tr><tr><td>4-faces</td><td>12</td><td>6 <a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a>6 <a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td></tr><tr><td>Cells</td><td>60</td><td>45 <a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a>15 <a href="/facts/Truncated_tetrahedron/K9ItvmUK">t{3,3}</a></td></tr><tr><td>Faces</td><td>140</td><td>80 <a href="/facts/Triangle/cRXUwsMn">{3}</a>60 <a href="/facts/Hexagon/DPzuCzWE">{6}</a></td></tr><tr><td>Edges</td><td colspan="2">150</td></tr><tr><td>Vertices</td><td colspan="2">60</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td colspan="2"><a href="/facts/5-cell/Rh6lwDcc">{ }v{3}</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td colspan="2">A5 [3,3,3,3], order 720</td></tr><tr><td>Properties</td><td colspan="2"><a href="/facts/Convex_polytope/2pRHcRgC">convex</a></td></tr></tbody></table>
<h3>Alternate names</h3>
<ul><li>Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul>
<h3>Coordinates</h3>
<p>The vertices of the <i>bitruncated 5-simplex</i> can be most simply constructed on a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> in 6-space as permutations of (0,0,0,1,2,2) <i>or</i> of (0,0,1,2,2,2). These represent positive <a href="/facts/Orthant/3bkJde4G">orthant</a> facets of the <a href="/facts/Bitruncated_6-orthoplex/xwdfvuzF">bitruncated 6-orthoplex</a>, and the <a href="/facts/Tritruncated_6-cube/uDq8zaRk">tritruncated 6-cube</a> respectively.
</p>
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th>Ak<a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>A5</th><th>A4</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[6]</td><td>[5]</td></tr><tr><th>Ak<a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>A3</th><th>A2</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[4]</td><td>[3]</td></tr></tbody></table>
<h2 id="related-uniform-5-polytopes">Related uniform 5-polytopes</h2>
<p>The truncated 5-simplex is one of 19 <a href="/facts/Uniform_5-polytope/mqxsoqwY">uniform 5-polytopes</a> based on the [3,3,3,3] <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a>, all shown here in A5 <a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a> <a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a>. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
</p>
<table><tbody><tr><th colspan="12">A5 polytopes</th></tr><tr><td><a href="/facts/5-simplex/KIfBFfhl">t0</a></td><td><a href="/facts/Rectified_5-simplex/H5kx0rFW">t1</a></td><td><a href="/facts/Birectified_5-simplex/H5kx0rFW">t2</a></td><td><a href="/facts/Truncated_5-simplex/GFrpum4c">t0,1</a></td><td><a href="/facts/Cantellated_5-simplex/httz7gDl">t0,2</a></td><td><a href="/facts/Bitruncated_5-simplex/GFrpum4c">t1,2</a></td><td><a href="/facts/Runcinated_5-simplex/tylUERyx">t0,3</a></td></tr><tr><td><a href="/facts/Bicantellated_5-simplex/httz7gDl">t1,3</a></td><td><a href="/facts/Stericated_5-simplex/eUVYy57D">t0,4</a></td><td><a href="/facts/Cantitruncated_5-simplex/httz7gDl">t0,1,2</a></td><td><a href="/facts/Runcitruncated_5-simplex/tylUERyx">t0,1,3</a></td><td><a href="/facts/Runcicantellated_5-simplex/tylUERyx">t0,2,3</a></td><td><a href="/facts/Bicantitruncated_5-simplex/httz7gDl">t1,2,3</a></td><td><a href="/facts/Steritruncated_5-simplex/eUVYy57D">t0,1,4</a></td></tr><tr><td><a href="/facts/Stericantellated_5-simplex/eUVYy57D">t0,2,4</a></td><td><a href="/facts/Runcicantitruncated_5-simplex/tylUERyx">t0,1,2,3</a></td><td><a href="/facts/Stericantitruncated_5-simplex/eUVYy57D">t0,1,2,4</a></td><td><a href="/facts/Steriruncitruncated_5-simplex/eUVYy57D">t0,1,3,4</a></td><td><a href="/facts/Omnitruncated_5-simplex/eUVYy57D">t0,1,2,3,4</a></td></tr></tbody></table>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a>:
<ul><li>H.S.M. Coxeter, <i>Regular Polytopes</i>, 3rd Edition, Dover New York, 1973</li>
<li>Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6 <a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[1]</a>
<ul><li>(Paper 22) H.S.M. Coxeter, <i>Regular and Semi Regular Polytopes I</i>, [Math. Zeit. 46 (1940) 380-407, MR 2,10]</li>
<li>(Paper 23) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes II</i>, [Math. Zeit. 188 (1985) 559-591]</li>
<li>(Paper 24) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes III</i>, [Math. Zeit. 200 (1988) 3-45]</li></ul></li></ul></li>
<li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> <i>Uniform Polytopes</i>, Manuscript (1991)
<ul><li>N.W. Johnson: <i>The Theory of Uniform Polytopes and Honeycombs</i>, Ph.D.</li></ul></li>
<li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/polytera.htm">"5D uniform polytopes (polytera)"</a>. x3x3o3o3o - tix, o3x3x3o3o - bittix</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#simplex">Glossary for hyperspace</a>, George Olshevsky.</li>
<li><a href="http://www.polytope.net/hedrondude/topes.htm">Polytopes of Various Dimensions</a>, Jonathan Bowers
<ul><li><a href="http://www.polytope.net/hedrondude/truncates5.htm">Truncated uniform polytera</a> (tix), Jonathan Bowers</li></ul></li>
<li><a href="http://tetraspace.alkaline.org/glossary.htm">Multi-dimensional Glossary</a></li></ul>
<table><tbody><tr><th colspan="13"><ul><li>v</li><li>t</li><li>e</li></ul>Fundamental convex <a href="/facts/Regular_polytope/RYTnEF6Q">regular</a> and <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytopes</a> in dimensions 2–10</th></tr><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Family</a></th><td><a href="/facts/Simple_Lie_group/LkHTNYU9"><i>A</i><i>n</i></a></td><td><a href="/facts/Simple_Lie_group/LkHTNYU9"><i>B</i><i>n</i></a></td><td><i>I</i>2(p) / <a href="/facts/Simple_Lie_group/LkHTNYU9"><i>D</i><i>n</i></a></td><td><a href="/facts/E6_(mathematics)/U9YEMtWj"><i>E</i>6</a> / <a href="/facts/E7_(mathematics)/KjFez9PW"><i>E</i>7</a> / <a href="/facts/E8_(mathematics)/NEaKT71H"><i>E</i>8</a> / <a href="/facts/F4_(mathematics)/30H8CQ8v"><i>F</i>4</a> / <a href="/facts/G2_(mathematics)/Bka4pKjQ"><i>G</i>2</a></td><td><a href="/facts/H4_(mathematics)/bgzHFm1Q">Hn</a></td></tr><tr><th><a href="/facts/Regular_polygon/ws89Nhpt">Regular polygon</a></th><td><a href="/facts/Equilateral_triangle/MJomsYDS">Triangle</a></td><td><a href="/facts/Square/JHJeMvmL">Square</a></td><td><a href="/facts/Regular_polygon/ws89Nhpt">p-gon</a></td><td><a href="/facts/Hexagon/DPzuCzWE">Hexagon</a></td><td><a href="/facts/Pentagon/jeHEARHH">Pentagon</a></td></tr><tr><th><a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform polyhedron</a></th><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a></td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a> • <a href="/facts/Cube/65lUaAqN">Cube</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">Demicube</a></td><td></td><td><a href="/facts/Regular_dodecahedron/xClYqhcp">Dodecahedron</a> • <a href="/facts/Regular_icosahedron/1U93vmXV">Icosahedron</a></td></tr><tr><th><a href="/facts/Uniform_polychoron/NHxomMxC">Uniform polychoron</a></th><td><a href="/facts/5-cell/Rh6lwDcc">Pentachoron</a></td><td><a href="/facts/16-cell/X0YAfobw">16-cell</a> • <a href="/facts/Tesseract/tlqouBoU">Tesseract</a></td><td><a href="/facts/16-cell/X0YAfobw">Demitesseract</a></td><td><a href="/facts/24-cell/tfoGnlAD">24-cell</a></td><td><a href="/facts/120-cell/W4nroBjJ">120-cell</a> • <a href="/facts/600-cell/tNBtDUe9">600-cell</a></td></tr><tr><th><a href="/facts/Uniform_5-polytope/mqxsoqwY">Uniform 5-polytope</a></th><td><a href="/facts/5-simplex/KIfBFfhl">5-simplex</a></td><td><a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a> • <a href="/facts/5-cube/XABdhjof">5-cube</a></td><td><a href="/facts/5-demicube/zvCeNbU2">5-demicube</a></td><td></td><td></td></tr><tr><th><a href="/facts/Uniform_6-polytope/pxqFTw2F">Uniform 6-polytope</a></th><td><a href="/facts/6-simplex/mInfSIeH">6-simplex</a></td><td><a href="/facts/6-orthoplex/eGV6o0ol">6-orthoplex</a> • <a href="/facts/6-cube/xN74tdD0">6-cube</a></td><td><a href="/facts/6-demicube/1oTufnFp">6-demicube</a></td><td><a href="/facts/1_22_polytope/7XXjWGQ8">122</a> • <a href="/facts/2_21_polytope/nF3nYaQk">221</a></td><td></td></tr><tr><th><a href="/facts/Uniform_7-polytope/suCJSvTN">Uniform 7-polytope</a></th><td><a href="/facts/7-simplex/6DwtPplH">7-simplex</a></td><td><a href="/facts/7-orthoplex/mIJa3rQp">7-orthoplex</a> • <a href="/facts/7-cube/4af1W7tq">7-cube</a></td><td><a href="/facts/7-demicube/fGVUTvTI">7-demicube</a></td><td><a href="/facts/1_32_polytope/um16Pd2V">132</a> • <a href="/facts/2_31_polytope/zkjkKJ6X">231</a> • <a href="/facts/3_21_polytope/AEUzwA1A">321</a></td><td></td></tr><tr><th><a href="/facts/Uniform_8-polytope/M93v3h3c">Uniform 8-polytope</a></th><td><a href="/facts/8-simplex/QUhcnmk2">8-simplex</a></td><td><a href="/facts/8-orthoplex/dXFkXtPn">8-orthoplex</a> • <a href="/facts/8-cube/vEpKH3le">8-cube</a></td><td><a href="/facts/8-demicube/sWAomIhw">8-demicube</a></td><td><a href="/facts/1_42_polytope/DTgYDddB">142</a> • <a href="/facts/2_41_polytope/R9z0wNUz">241</a> • <a href="/facts/4_21_polytope/QFGglCkk">421</a></td><td></td></tr><tr><th><a href="/facts/Uniform_9-polytope/UdFL9ns2">Uniform 9-polytope</a></th><td><a href="/facts/9-simplex/dHV7qhxN">9-simplex</a></td><td><a href="/facts/9-orthoplex/D7LjUcAh">9-orthoplex</a> • <a href="/facts/9-cube/RNgCPzoM">9-cube</a></td><td><a href="/facts/9-demicube/dkTpfqss">9-demicube</a></td><td></td><td></td></tr><tr><th><a href="/facts/Uniform_10-polytope/Qc5NglV8">Uniform 10-polytope</a></th><td><a href="/facts/10-simplex/oeft7Qjn">10-simplex</a></td><td><a href="/facts/10-orthoplex/x6ysTC9U">10-orthoplex</a> • <a href="/facts/10-cube/mmmyBfJJ">10-cube</a></td><td><a href="/facts/10-demicube/TskxfLy9">10-demicube</a></td><td></td><td></td></tr><tr><th>Uniform <i>n</i>-<a href="/facts/Polytope/a76nXrL0">polytope</a></th><td><i>n</i>-<a href="/facts/Simplex/0FCfNRN8">simplex</a></td><td><i>n</i>-<a href="/facts/Cross-polytope/erk0byF6">orthoplex</a> • <i>n</i>-<a href="/facts/Hypercube/q2mRPbIz">cube</a></td><td><i>n</i>-<a href="/facts/Demihypercube/ackHKsEa">demicube</a></td><td><a href="/facts/Uniform_1_k2_polytope/eqTstjgl">1k2</a> • <a href="/facts/Uniform_2_k1_polytope/pQv1I7AC">2k1</a> • <a href="/facts/Uniform_k_21_polytope/vv3lWMYd">k21</a></td><td><i>n</i>-<a href="/facts/Pentagonal_polytope/4uxePdKt">pentagonal polytope</a></td></tr><tr><th colspan="13">Topics: <a href="/facts/Polytope_families/Gun0RgmR">Polytope families</a> • <a href="/facts/Regular_polytope/RYTnEF6Q">Regular polytope</a> • <a href="/facts/List_of_regular_polytopes_and_compounds/Kng25Ymq">List of regular polytopes and compounds</a></th></tr></tbody></table>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Klitizing, (x3x3o3o3o - tix) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Klitizing, (o3x3x3o3o - bittix) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Truncated 5-simplexes open-in-new

Truncated 5-simplexes