Truncated 7-cubes

<h2 id="truncated-7-cube">Truncated 7-cube</h2>
<table><tbody><tr><th colspan="2">Truncated 7-cube</th></tr><tr><td>Type</td><td><a href="/facts/Uniform_7-polytope/suCJSvTN">uniform 7-polytope</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>t{4,35}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a></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>3136</td></tr><tr><td>Vertices</td><td>896</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>Elongated 5-simplex pyramid</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td>B7, [35,4]</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>Truncated hepteract (Jonathan Bowers)<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul>
<h3>Coordinates</h3>
<p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate <a href="/facts/Permutation/JSw9ukmv">permutations</a> of
</p>
(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th><a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>B7 / A6</th><th>B6 / D7</th><th>B5 / D6 / A4</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[14]</td><td>[12]</td><td>[10]</td></tr><tr><th>Coxeter plane</th><th>B4 / D5</th><th>B3 / D4 / A2</th><th>B2 / D3</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[8]</td><td>[6]</td><td>[4]</td></tr><tr><th>Coxeter plane</th><th>A5</th><th>A3</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[6]</td><td>[4]</td></tr></tbody></table>
<h3>Related polytopes</h3>
<p>The <i><a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> <a href="/facts/7-cube/4af1W7tq">7-cube</a></i>, is sixth in a sequence of truncated <a href="/facts/Hypercube/q2mRPbIz">hypercubes</a>:
</p>
Truncated hypercubes<table><tbody><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan="3">...</td></tr><tr><th>Name</th><td><a href="/facts/Octagon/H5PuP3wP">Octagon</a></td><td><a href="/facts/Truncated_cube/XVAUG4fD">Truncated cube</a></td><td><a href="/facts/Truncated_tesseract/ShQzMKoa">Truncated tesseract</a></td><td><a href="/facts/Truncated_5-cube/kLk4Dwvl">Truncated 5-cube</a></td><td><a href="/facts/Truncated_6-cube/uDq8zaRk">Truncated 6-cube</a></td><td><a href="/facts/Truncated_7-cube/M0SeeGkR">Truncated 7-cube</a></td><td><a href="/facts/Truncated_8-cube/8Bsbj6Nw">Truncated 8-cube</a></td></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></th><td>( )v( )</td><td><a href="/facts/Isosceles_triangle/rvLe3yqK">( )v{ }</a></td><td><a href="/facts/Triangular_pyramid/7mkSrl6j">( )v{3}</a></td><td><a href="/facts/5-cell/Rh6lwDcc">( )v{3,3}</a></td><td><a href="/facts/5-simplex/KIfBFfhl">( )v{3,3,3}</a></td><td><a href="/facts/6-simplex/mInfSIeH">( )v{3,3,3,3}</a></td><td><a href="/facts/7-simplex/6DwtPplH">( )v{3,3,3,3,3}</a></td></tr></tbody></table>
<h2 id="bitruncated-7-cube">Bitruncated 7-cube</h2>
<table><tbody><tr><th colspan="2">Bitruncated 7-cube</th></tr><tr><td>Type</td><td><a href="/facts/Uniform_7-polytope/suCJSvTN">uniform 7-polytope</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>2t{4,35}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a></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>9408</td></tr><tr><td>Vertices</td><td>2688</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>{ }v{3,3,3}</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td>B7, [35,4]D7, [34,1,1]</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>Bitruncated hepteract (Jonathan Bowers)<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul>
<h3>Coordinates</h3>
<p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate <a href="/facts/Permutation/JSw9ukmv">permutations</a> of
</p>
(±2,±2,±2,±2,±2,±1,0)
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th><a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>B7 / A6</th><th>B6 / D7</th><th>B5 / D6 / A4</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[14]</td><td>[12]</td><td>[10]</td></tr><tr><th>Coxeter plane</th><th>B4 / D5</th><th>B3 / D4 / A2</th><th>B2 / D3</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[8]</td><td>[6]</td><td>[4]</td></tr><tr><th>Coxeter plane</th><th>A5</th><th>A3</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[6]</td><td>[4]</td></tr></tbody></table>
<h3>Related polytopes</h3>
<p>The <i><a href="/facts/Bitruncation/t0xarbyJ">bitruncated</a> <a href="/facts/7-cube/4af1W7tq">7-cube</a></i> is fifth in a sequence of bitruncated <a href="/facts/Hypercube/q2mRPbIz">hypercubes</a>:
</p>
Bitruncated hypercubes<table><tbody><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan="3">...</td></tr><tr><th>Name</th><td><a href="/facts/Truncated_octahedron/GA98b4TH">Bitruncated cube</a></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">Bitruncated tesseract</a></td><td><a href="/facts/Bitruncated_5-cube/kLk4Dwvl">Bitruncated 5-cube</a></td><td><a href="/facts/Bitruncated_6-cube/uDq8zaRk">Bitruncated 6-cube</a></td><td><a href="/facts/Bitruncated_7-cube/M0SeeGkR">Bitruncated 7-cube</a></td><td><a href="/facts/Bitruncated_8-cube/8Bsbj6Nw">Bitruncated 8-cube</a></td></tr><tr><th>Coxeter</th><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Vertex figure</th><td><a href="/facts/Isosceles_triangle/rvLe3yqK">( )v{ }</a></td><td><a href="/facts/Disphenoid/7rU7jc7U">{ }v{ }</a></td><td><a href="/facts/5-cell/Rh6lwDcc">{ }v{3}</a></td><td><a href="/facts/5-simplex/KIfBFfhl">{ }v{3,3}</a></td><td><a href="/facts/6-simplex/mInfSIeH">{ }v{3,3,3}</a></td><td><a href="/facts/7-simplex/6DwtPplH">{ }v{3,3,3,3}</a></td></tr></tbody></table>
<h2 id="tritruncated-7-cube">Tritruncated 7-cube</h2>
<table><tbody><tr><th colspan="2">Tritruncated 7-cube</th></tr><tr><td>Type</td><td><a href="/facts/Uniform_7-polytope/suCJSvTN">uniform 7-polytope</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>3t{4,35}</td></tr><tr><td><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a></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>13440</td></tr><tr><td>Vertices</td><td>3360</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>{4}v{3,3}</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td>B7, [35,4]D7, [34,1,1]</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>Tritruncated hepteract (Jonathan Bowers)<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li></ul>
<h3>Coordinates</h3>
<p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate <a href="/facts/Permutation/JSw9ukmv">permutations</a> of
</p>
(±2,±2,±2,±2,±1,0,0)
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th><a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>B7 / A6</th><th>B6 / D7</th><th>B5 / D6 / A4</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[14]</td><td>[12]</td><td>[10]</td></tr><tr><th>Coxeter plane</th><th>B4 / D5</th><th>B3 / D4 / A2</th><th>B2 / D3</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[8]</td><td>[6]</td><td>[4]</td></tr><tr><th>Coxeter plane</th><th>A5</th><th>A3</th></tr><tr><th>Graph</th><td></td><td></td></tr><tr><th>Dihedral symmetry</th><td>[6]</td><td>[4]</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/polyexa.htm">"7D uniform polytopes (polyexa)"</a>. o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.polytope.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>Klitizing (x3x3o3o3o3o4o - taz) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Klitizing (o3x3x3o3o3o4o - botaz) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Klitizing (o3o3x3x3o3o4o - totaz) <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Truncated 7-cubes open-in-new

Truncated 7-cubes