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Heptellated 8-simplexes
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<h2 id="heptellated-8-simplex">Heptellated 8-simplex</h2> <table><tbody><tr><th colspan="2">Heptellated 8-simplex</th></tr><tr><td>Type</td><td><a href="/facts/Uniform_8-polytope/M93v3h3c">uniform 8-polytope</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,7{3,3,3,3,3,3,3}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a></td><td></td></tr><tr><td>7-faces</td><td></td></tr><tr><td>6-faces</td><td></td></tr><tr><td>5-faces</td><td></td></tr><tr><td>4-faces</td><td></td></tr><tr><td>Cells</td><td></td></tr><tr><td>Faces</td><td></td></tr><tr><td>Edges</td><td>504</td></tr><tr><td>Vertices</td><td>72</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>6-simplex antiprism</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td>A8×2, [[37]], order 725760</td></tr><tr><td>Properties</td><td><a href="/facts/Convex_polytope/2pRHcRgC">convex</a></td></tr></tbody></table> <h3>Alternate names</h3> <ul><li>Expanded 8-simplex</li> <li>Small exated enneazetton (soxeb) (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>heptellated 8-simplex</i> can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on <a href="/facts/Facet_(geometry)/IreRHGJ7">facets</a> of the heptellated 9-orthoplex. </p><p>A second construction in 9-space, from the center of a <a href="/facts/Rectified_9-orthoplex/EN3Qax3H">rectified 9-orthoplex</a> is given by coordinate permutations of: </p> (1,-1,0,0,0,0,0,0,0) <h3>Root vectors</h3> <p>Its 72 vertices represent the root vectors of the <a href="/facts/Simple_Lie_group/LkHTNYU9">simple Lie group</a> A8. </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>A8</th><th>A7</th><th>A6</th><th>A5</th></tr><tr><th>Graph</th><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[[9]] = [18]</td><td>[8]</td><td>[[7]] = [14]</td><td>[6]</td></tr><tr><th>Ak Coxeter plane</th><th>A4</th><th>A3</th><th>A2</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[[5]] = [10]</td><td>[4]</td><td>[[3]] = [6]</td></tr></tbody></table> <h2 id="omnitruncated-8-simplex">Omnitruncated 8-simplex</h2> <table><tbody><tr><th colspan="2">Omnitruncated 8-simplex</th></tr><tr><td>Type</td><td><a href="/facts/Uniform_8-polytope/M93v3h3c">uniform 8-polytope</a></td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t0,1,2,3,4,5,6,7{37}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a></td><td></td></tr><tr><td>7-faces</td><td></td></tr><tr><td>6-faces</td><td></td></tr><tr><td>5-faces</td><td></td></tr><tr><td>4-faces</td><td></td></tr><tr><td>Cells</td><td></td></tr><tr><td>Faces</td><td></td></tr><tr><td>Edges</td><td>1451520</td></tr><tr><td>Vertices</td><td>362880</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>irr. 7-simplex</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></td><td>A8, [[37]], order 725760</td></tr><tr><td>Properties</td><td><a href="/facts/Convex_polytope/2pRHcRgC">convex</a></td></tr></tbody></table> <p>The symmetry order of an omnitruncated 8-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the <a href="/facts/Omnitruncation/hmbMrRm6">omnitruncation</a>, being 362880 (9 <a href="/facts/Factorial/kkfy1Bwe">factorial</a>) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual. </p> <h3>Alternate names</h3> <ul><li>Heptihexipentisteriruncicantitruncated 8-simplex</li> <li>Great exated enneazetton (goxeb) (Jonathan Bowers)<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul> <h3>Coordinates</h3> <p>The <a href="/facts/Cartesian_coordinate/lkTqvMmB">Cartesian coordinates</a> of the vertices of the <i>omnitruncated 8-simplex</i> can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on <a href="/facts/Facet_(geometry)/IreRHGJ7">facets</a> of the heptihexipentisteriruncicantitruncated 9-orthoplex, t0,1,2,3,4,5,6,7{37,4} </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>A8</th><th>A7</th><th>A6</th><th>A5</th></tr><tr><th>Graph</th><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[[9]] = [18]</td><td>[8]</td><td>[[7]] = [14]</td><td>[6]</td></tr><tr><th>Ak Coxeter plane</th><th>A4</th><th>A3</th><th>A2</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[[5]] = [10]</td><td>[4]</td><td>[[3]] = [6]</td></tr></tbody></table> <h3>Permutohedron and related tessellation</h3> <p>The <i>omnitruncated 8-simplex</i> is the <a href="/facts/Permutohedron/No1KH0rE">permutohedron</a> of order 9. The omnitruncated 8-simplex is a <a href="/facts/Zonotope/w16kH9H3">zonotope</a>, the <a href="/facts/Minkowski_sum/dEQfoJ1A">Minkowski sum</a> of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex. </p><p>Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can <a href="/facts/Tessellate/7BWDywXC">tessellate</a> space by itself, in this case 8-dimensional space with three facets around each <a href="/facts/Ridge_(geometry)/T2E6BTU4">ridge</a>. It has <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a> of . </p> <h2 id="related-polytopes">Related polytopes</h2> <p>The two presented polytopes are selected from 135 <a href="/facts/8-polytope/M93v3h3c">uniform 8-polytopes</a> with A8 symmetry, shown in the table below. </p> <table><tbody><tr><th colspan="12">A8 polytopes</th></tr><tr><td><a href="/facts/8-simplex/QUhcnmk2">t0</a></td><td><a href="/facts/Rectified_8-simplex/SozS9o74">t1</a></td><td><a href="/facts/Birectified_8-simplex/SozS9o74">t2</a></td><td><a href="/facts/Trirectified_8-simplex/SozS9o74">t3</a></td><td><a href="/facts/Truncated_8-simplex/aQ2TjRuJ">t01</a></td><td><a href="/facts/Cantellated_8-simplex/BBMyy5Gm">t02</a></td><td><a href="/facts/Bitruncated_8-simplex/aQ2TjRuJ">t12</a></td><td><a href="/facts/Runcinated_8-simplex/eucoYK9b">t03</a></td><td><a href="/facts/Bicantellated_8-simplex/BBMyy5Gm">t13</a></td><td><a href="/facts/Tritruncated_8-simplex/aQ2TjRuJ">t23</a></td><td><a href="/facts/Stericated_8-simplex/5YrBlfyj">t04</a></td><td><a href="/facts/Biruncinated_8-simplex/eucoYK9b">t14</a></td><td><a href="/facts/Tricantellated_8-simplex/BBMyy5Gm">t24</a></td><td><a href="/facts/Quadritruncated_8-simplex/aQ2TjRuJ">t34</a></td><td><a href="/facts/Pentellated_8-simplex/Wd4EzKld">t05</a></td></tr><tr><td><a href="/facts/Bistericated_8-simplex/5YrBlfyj">t15</a></td><td><a href="/facts/Triruncinated_8-simplex/eucoYK9b">t25</a></td><td><a href="/facts/Hexicated_8-simplex/30whwsV8">t06</a></td><td><a href="/facts/Bipentellated_8-simplex/Wd4EzKld">t16</a></td><td><a href="/facts/Heptellated_8-simplex/VX5EgAv7">t07</a></td><td><a href="/facts/Cantitruncated_8-simplex/BBMyy5Gm">t012</a></td><td><a href="/facts/Runcitruncated_8-simplex/eucoYK9b">t013</a></td><td><a href="/facts/Runcicantellated_8-simplex/eucoYK9b">t023</a></td><td><a href="/facts/Bicantitruncated_8-simplex/BBMyy5Gm">t123</a></td><td><a href="/facts/Steritruncated_8-simplex/5YrBlfyj">t014</a></td><td><a href="/facts/Stericantellated_8-simplex/5YrBlfyj">t024</a></td><td><a href="/facts/Biruncitruncated_8-simplex/eucoYK9b">t124</a></td><td><a href="/facts/Steriruncinated_8-simplex/5YrBlfyj">t034</a></td><td><a href="/facts/Biruncicantellated_8-simplex/eucoYK9b">t134</a></td><td><a href="/facts/Tricantitruncated_8-simplex/BBMyy5Gm">t234</a></td></tr><tr><td>t015</td><td>t025</td><td><a href="/facts/Bisteritruncated_8-simplex/5YrBlfyj">t125</a></td><td>t035</td><td><a href="/facts/Bistericantellated_8-simplex/5YrBlfyj">t135</a></td><td>t235</td><td>t045</td><td><a href="/facts/Bisteriruncinated_8-simplex/5YrBlfyj">t145</a></td><td>t016</td><td>t026</td><td>t126</td><td>t036</td><td>t136</td><td>t046</td><td>t056</td></tr><tr><td>t017</td><td>t027</td><td>t037</td><td><a href="/facts/Runcicantitruncated_8-simplex/eucoYK9b">t0123</a></td><td><a href="/facts/Stericantitruncated_8-simplex/5YrBlfyj">t0124</a></td><td><a href="/facts/Steriruncitruncated_8-simplex/5YrBlfyj">t0134</a></td><td><a href="/facts/Steriruncicantellated_8-simplex/5YrBlfyj">t0234</a></td><td><a href="/facts/Biruncicantitruncated_8-simplex/eucoYK9b">t1234</a></td><td>t0125</td><td>t0135</td><td>t0235</td><td><a href="/facts/Bistericantitruncated_8-simplex/5YrBlfyj">t1235</a></td><td>t0145</td><td>t0245</td><td><a href="/facts/Bisteriruncitruncated_8-simplex/5YrBlfyj">t1245</a></td></tr><tr><td>t0345</td><td><a href="/facts/Bisteriruncicantellated_8-simplex/5YrBlfyj">t1345</a></td><td><a href="/facts/Triruncicantitruncated_8-simplex/eucoYK9b">t2345</a></td><td>t0126</td><td>t0136</td><td>t0236</td><td>t1236</td><td>t0146</td><td>t0246</td><td>t1246</td><td>t0346</td><td>t1346</td><td>t0156</td><td>t0256</td><td>t1256</td></tr><tr><td>t0356</td><td>t0456</td><td>t0127</td><td>t0137</td><td>t0237</td><td>t0147</td><td>t0247</td><td>t0347</td><td>t0157</td><td>t0257</td><td>t0167</td><td><a href="/facts/Steriruncicantitruncated_8-simplex/5YrBlfyj">t01234</a></td><td>t01235</td><td>t01245</td><td>t01345</td></tr><tr><td>t02345</td><td><a href="/facts/Bisteriruncicantitruncated_8-simplex/5YrBlfyj">t12345</a></td><td>t01236</td><td>t01246</td><td>t01346</td><td>t02346</td><td>t12346</td><td>t01256</td><td>t01356</td><td>t02356</td><td>t12356</td><td>t01456</td><td>t02456</td><td>t03456</td><td>t01237</td></tr><tr><td>t01247</td><td>t01347</td><td>t02347</td><td>t01257</td><td>t01357</td><td>t02357</td><td>t01457</td><td>t01267</td><td>t01367</td><td>t012345</td><td>t012346</td><td>t012356</td><td>t012456</td><td>t013456</td><td>t023456</td></tr><tr><td>t123456</td><td>t012347</td><td>t012357</td><td>t012457</td><td>t013457</td><td>t023457</td><td>t012367</td><td>t012467</td><td>t013467</td><td>t012567</td><td>t0123456</td><td>t0123457</td><td>t0123467</td><td>t0123567</td><td><a href="/facts/Omnitruncated_8-simplex/VX5EgAv7">t01234567</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/polyzetta.htm">"8D uniform polytopes (polyzetta)"</a>. x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm">Polytopes of Various Dimensions</a></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>Klitzing, (x3o3o3o3o3o3o3x - soxeb) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Klitzing, (x3x3x3x3x3x3x3x - goxeb) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>